






'^■^^^;^'r 



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CYCLOPiEDIA OF DRAWING, 



DESIGNED AS 



J^ TEXT-BOOK 



FOE THE 



MECHANIC, ARCHITECT, ENGINEEE, AND SURVEYOR. 

COMPEISING 

GEOMETRICAL PROJECTION, MECHANICAL, ARCHITECTURAL, AND TOPOGRAPHICAL 
DRAWING, PERSPECTIYE AND ISOMETRY. 



EDITED BY 



W. E. AVOHTHEN 



:^EW rOEK: 
D. APPLET OX AXD COMPACT, 

443&445B ROADWAY. 
1866. 



5^ 






Entered, according to Act of Congress, in the year 1857, by 

D. APPLETON & CO., 

In the Clerk's OilSce of the District Court of the United States for the Southern 

District of New York. 






PREFACE 



At the suggestion of the piibhshers this work was undertaken to 
form one of their series of Dictionaries and Cyclopaedias/ In this 
view, it has been the intention to make it a complete com^se of instruc- 
tion and book of reference to the mechanic, architect and engineer. 
It has not therefore been confined to the explanation and illustration 
of the methods of projection, and the delineation of objects which 
might serve as copies to the draughtsman, matters of essential impor- 
tance for the correct and intelligible representation of every form ; but 
it contains the means of determining the amount and direction of 
strains, to which different parts of a machine or structure may be sub- 
jected, and the rules for disposing and proportioning of the material 
employed, to the safe and permanent resistance of those strains, with 
practical applications of the same. Thus while it supplies numerous 
illustrations in every department for the mere copyistj^^lt also affords 
suggestions and aids to the mechanic in the execution of new designs. 
And although the arranging and properly proportioning alone of mate- 
rial in a suitable direction and adequately to the resistance of the strains 
to which it might be exposed, would produce a structm^e sufficient in 
point of strength for the purposes for which it is intended, yet as in 
many cases the disposition of the material may be applied not only 
practically, but also artistically, and adapted to the reception of orna- 
ment, under the head of Architectural Dramng the general charac- 
teristics of various styles have been treated of, and illustrated, with 
brief remarks on proportion and the application of color. 



IV PREFACE. 

Within the last few years both here and abroad, a number of works 
have been pubhshed on " Practical Drawing," but no one work has il- 
lustrated all departments of the subject. In the mechanical, the works 
cf M. Le Brun and M. M. Armengaud are the standard which have 
been made the basis of two English works, " The practical Draughts- 
man's Book of Industrial Design " and the " Engineer's and Machinist's 
Drawing Book." Erom the latter of these works we have drawn most 
of our chapters on Geometrical and Mechanical Drawing, and Shades 
and Shadows. In neither the Erench nor the English works has the 
science of architectural construction and drawing been adequately illus- 
trated, nor has Topographical Drawing been treated of I*ythese two 
departments a varied selection has been made from the best authorities. 
In the Architectural, Eerguson and Garbett have be on the most con- 
sulted ; in the Topographical, Williams, Gillespie, Smith and Erome. 
The work Avill be found quite fully illustrated, and the drawings and 
engravings have been carefully executed, mostly under the supervision 
of Mr. H. Grassau. 

Like most cyclopaedias, this work claims for its articles but little of 
novelty or originality; the intention of the compiler was to collect 
^vithin moderate compass as much valuable matter as possible, in prac- 
tical Drawing and Design; and to this piu^pose he brings the expe- 
rience of series of years in each of the departments treated. Practi- 
cally, he has had means of knowing the necessities of the trade and of 
the profession, and trusts that the selection nov\^ made will be found 
usefid for the purposes for which it was intended. — W. 



/' 



TABLE OF CONTENTS. 



OEOMETEICAL DEFIXITIOXS OF TECHXICALITIES. 

Point defined, 1 ; given points defined, 1 ; 
lines defined, 1 ; parallel lines defined, 1 ; 
horizontal lines defined, 1 ; superficies de- 
scribed, 1 ; solids described, 1 ; lines verti- 
cal defined, 1 : inclined lines defined, 1 ; an- 
gle defined, 2 ; obtuse angle defined, 2 ; 
acute angle defined, 2 ; triangle defined, 2 ; 
equilateral triangle, 2; isosceles triangle, 

2 : scalene triangle, 2 ; quadrilateral figure 
defined, 2 ; parallelogram described, 2 ; 
square described, 3; rectangle described, 

3 ; rhombus described, 3 ; trapezium de- 
scribed, 3 ; polygons described, 3 ; regular 
polygons described, 3 ; circle described, 3 
concentric, 3 ; eccentric, 3 ; arc defined, 3 

. sector defined, 3 j chord defined, 3 ; circle 
its circumference divided, 4 ; ellipse defined, 
4 ;. parabola described, 4 ; hyperbola de- 
scribed, 5 ; cycloid described, 5 ; epic}"- 
cloid described, 5 ; liA-pocycloid described, 5. 

OF SOLIDS. 

Prism described, 5 ; pyramid described, 5 : 
sphere defined, 6 ; cylinder defined, G ; cone 
defined, 6 ; tetrahedron described, 6 ; hex- 
ahedron described. G ; octahedron described, 
6 ; dodecahedron described, G ; isosahedron 
described, G. 

DEATVIXG IXSTErMEXTS. 

Ruler common, 7 ; hovr made, 7 ; uses, 7 ; par- 
allel ruler, 10 ; hoTV made, 10 ; triangle 
wooden, 7 ; how made, 7 ; uses, 8 ; square, 
the T, 9 ; how made, 9 ; sweeps or vari- 
able curves, 10 ; compasses or dividers, 
11; hair dividers, 12; dividers with mov- 



able points. 12 ; bow dividers. 13 ; spring 
dividers, 13; drawing pen, 14; dotting 
point, 15 ; drawing pins, 15. 

Scales, 16. On the selection of the scale, 17 ; di- 

• '^agonal scales, 18 ; plotting scales and rulers. 
19 ; application of the protractor, 19 ; Ver- 
nier's scale, 19 ; general rule for Vernier's 
scale, 20 ; to set off an angle. 21 ; use of the 
lines of sines, secants, tangents and semi- 
tangents, 21 ; the line of rhombs, 21 ; the 
Ime of longitudes, 21 ; the sector, 22 ; plain 
scales on the sector, 22 ; sectoral double 
scales, 22 ; the line of lines, 23 ; to divide a 
given line into eight equal parts, 23 ; to form 
any required scale of equal parts, 23 ; ex- 
ample, 23 ; to construct a scale of feet and 
inches representing various dimensions, 23 ; 
examples, 23 ; line of chords, 23 ; line of 
polygons, 24 ; line of sines, tangents, &c., 
25 ; examples, 25 ; Marquois scales, 26 ; ex- 
ample, 2G. 

Trian(jvlar Compasses. 27 . Uses, 27; wholes 
'and halves, 28 ; proportional compasses, 28 ; 
beam compasses, 29 ; method of use, 29 ; 
portable or turn-in compasses, 30 ; tubular 
compasses of Brunei, 31 ; large screw di- 
viders, 31 ; circular protractor, 31 ; use, 32 ; 
pentagraph, 33 ; method of use, 34 ; camera 
lucida, 35 ; drawing table and board, 35. 

Drawing paper, 35 ; tracing paper, 36 ; smooth 
glue. 36 ; damp stretching, 37 ; mounting 
paper and drawings, 38 ; varnishing, 38 ; 
management of instruments, 39, 

GEOMETPJCAL TEOBLEMS. 

To draw straight lines through given points. 
42 ; how lines are divided in drawing, 42 ; 



VI 



TABLE OF CONTENT 



given line, 43-44-45 ; to draw a perpendic- 
ular to a straight line, 45 ; various methods, 
4o-4G ; to draw a straight line parallel to a 
given line, 47 ; to draw a parallel through a 
given point, 47 ; to construct an angle 
equal to a given angle, 48 ; to divide an an- 
gle, 48 ', to bisect an angle, 49 ; to describe 
an arc through given points with a given 
radius, 49 ; to find the centre of a given 
circle, 49 ; to describe a circle through 
three given points, 50; methods, 50; to 
draw a tangent to a circle from a given point 
in the circumference, 51 ; to draw a tangent 
to a circle from a point without it, 51 ; to 
describe a circle from a given point to touch 
a given circle, 52 ; to draw tangents to two 
given circles, 52 ; methods, 52 ; between 
two inclined lines to draw a series of circles 
touching each other and those lines, 53 ; 
between two inclined lines to draw a circu- 
lar segment to fill up the angle and touch- 
ing the line, 53 ; to fill up the angle of a 
straight line and a circle with a circular 
arch, &c., 54 ; to fill up the angle of a 
straight line and a circle with a circular 
arch, and to join the circle at a given point, 
54; to describe a circular arc joining two 
circles, and to touch one of them at a given 
point, 54 ; to find an arc tangent to a given 
j)oint on a straight line, 55 ; to connect 
two parallel lines by a reversed curve com- 
posed of two arcs of equal radius, 55 ; to 
join two given points in two given para:llel 
lines, by a reversed curve of two equal arcs, 
&c., 55. 
ProUems on Circles and Rectilinear Figures, 
56. To construct a triangle upon a given 
straight line, the length of the two sides 
being given, 50 ; to construct a square on a 
given line, 57 ; to construct a parallelogram 



on a given line. 



to describe a circle 



about a parallelogram, 58 ; to inscribe a circle 
in a triangle, 58 ; to inscribe a square in a 
circle, 58 ; to inscribe a circle in a square, 
59 ; to inscribe a pentagon in a circle, 59 ; 
to inscribe a hexagon in a circle, 59 ; to 
describe a hexagon about a circle, GO ; to 
construct a regular octagon on a given 
straight line, 00 ; to convert a square into 
a regular octagon, GO ; to inscribe an octa- 
gon in a circle, Gl ; to inscribe a circle 
within a polygon, Gl ; to describe a regu- 
lar octagon a})()Ut a circle, Gl ; to descri])c 
a circle witliout a polj-gon, Gl ; tabic of 
])()l3'g(>nal angles, G2 ; to inscribe a regu- 
lar polygon in a given circle, G2 ; to use the 



T square and triangle in the construction 
of some of the foregoing problems, 63, 

Sim2:)le application of regular Jig ures^ 65. To 
cover a surface with equilateral triangles, 
hexagons and lozenges, 65 ; to cover a sur- 
face with octagons and squares, 65. 

Prollems on proijortional lines and equiva- 
lent figures^ 66. To divide a straight line 
into two parts proportional to two given 
lines, 66 ; to divide a straight line into any 
number of parts of given proportions, ^^ ; 
to find a fourth proportional to three given 
lines, 67; to find a mean proportional to 
two, 67; to construct a triangle equal in 
area to a given rectangle, 67 ; to construct 
a square equal to a given rectangle, 67 ; to 
construct a triangle equivalent to any reg- 
ular polygon, 67. 

Problems on tlie Ellipse^ tTie Parabola, tlie 

^^' Hyperbola, the Cycloid and tlie Epicycloid, 
68. To describe an ellipse, the length and 
breadth being given, 68; methods, 68-69; 
to draw a tangent to a-n ellipse through a 
given point in the curve, 70 ; to draw a tan- 
gent to an ellipse from a given point with- 
out the curve, 70; to describe an ellipse 
approximately by means of circular arcs, 71. 

To construct a parabola when the focus and 
directrix are given, 72; methods, 72; to 
construct a parabola when other points are 
given, 73 ; to draw a tangent to a given 
point of the parabola, 73. 

To describe an hyperbola, 74 ; to draw a tan- 
gent to any point of an hj'pcrbola, 74. 

To describe a cycloid, 75 ; to describe an epi- 
cycloid, 76 ; methods, 77 ; to describe an 
involute, 78 ; to describe a spiral, 69. 

GEOMETEICAL TEOJECTIOX. 

Arithmetical and mechanical drawing de- 
scribed, 80. Shade Lines, 83 ; outline draw- 
ings, 83 ; French s^^stem of shading, 85. 

Projections of sim2^le hodioi, 85. Projection of 
an hexagonal pyramid, ?^^ ; projection of an 
hexagonal p3'ramid with the base in an in- 
clined position, &c., 86 ; to find the hori- 
zontal projection of a transveree section of 
said pyramid, 87 ; to find the horizontal 
projection of a transverse section of a reg- 
ular five-sided pyramid, 88. 

Projections of <i Prism, SS. To represent in 
plan and elevation a six-sided prism in an 
ujnight position, 88; to form the projec- 
tions of the same prism, 88 ; the in-ojeetions 
of the same, set in a })osition inclined to 
both planes of projection, 89. 



TABLE OF CONTENTS. 



Vll 



Construction of the Conic Sections^ 90; to find 
the horizontal projection of sections of a 
cone, 90-91 ; to find vertical projection of 
sections of two opposite cones, &c., 91. 
Penetrations or intersections of solids, 92. 
Penetrations of cylinders^ 92 ; explanations 
of figures, 93. Penetrations of cylinders^ 
cones and splieres^ 93 ; to find the curves 
resulting from the intersection of two cyl- 
inders, &c., 93 ; to find the curves of pen- 
etration in the elevation without the aid of 
the plan, 94 ; to find the curve of penetra- 
tion of a cone and sphere, 94 ; to find the 
curve of penetration of a cj'linder, and a 
cylindrical ring, 95 ; other problems, 96. 

Penetrations of cylinders, prisms, spheres and 
cones, 97. To delineate the lines of penetration 
of a sphere and a regular hexagonal prism, 
&c., 97 ; to delineate the lines of penetration 
of a c^dinder and a sphere, the centre of the 
sphere without the axis of the cylinder, 
98 ; to delineate the fines of penetration of 
a truncated cone and a prism, 98 ; to de- 
scribe the curves formed by the intersec- 
tion of a cylinder with the frustum of a 
cone, &c., 99. The Jielix, 99 ; problems, 100. 

DEVELOPMEXT OF SUEFACES. 

To develope the surface of a cylinder formed 
by the intersection of another cylinder, 102 ; 
to develope the surface of a frustum of a 
cone, 103 ; to develope the surface of a 
sphere or ball, 103. 

MECHANICS. 

Force defined, 105 ; direction of, 105 ; fines 
formed by three forces, 100; forces repre- 
sented by the sides of a polygon, 107; par- 
allel forces, 108 ; centre of graAdty, 109 ; to 
ascertain its position, 109. 

The Mechanical poicers, 110; the incline, 110. 
To find the power that will support a given 
weight, 111 ; to find the weight that will 
be sustained by a known power, 111 ; to 
find the height of an inclined plane neces- 
sary to sustain a given weight. 111 ; the 
wedge, 112; the screw, 112; to calculate 
the power imparted to a screw, 112; to 
find the weight raised, 112. Levers, 112; 
levers first class, 112 ; levers second class, 
113; levers third class, 113; to find the 
position of the fulcrum to support a given 
weight &c., 113 ; to find the weight &c., 
113; to find the power &c., 113. The 
wheel and axle, 114. The pulley, 114. 
Friction and limiting angle of resistance, 



115 ; experiments on friction by M. Morin, 
IIG. Of the equilihrium of the polygon of 
rods or cords, 117 ; methods of giving rigid- 
ity to a system of rods, 119. The mechan- 
ical 'properties of materials, 120; forces to 
which materials are subject, 120 ; pressure 
upon wooden posts, 120 ; pressure upon cast 
iron posts, 121 ; diameters to the 3.G power, 
122; lengths to the 1.7 power, 122; tensile 
strength of materials pulled in the direction 
of their length, 123. Transierse strength 
of materials, 124; examples, 124, 125 ; ex- 
periments on the transverse strength of rec- 
tangular tubes of wrought iron, &c., 127 ; 
approximate formula for riveted tubes, 127; 
detrusion, 128 ; torsion, 128 ; examples, 129. 
Mechanical tcorh or effect, 130; explained, 
130; unit of, 130; motors, 131; circum- 
stances demanding attention on the appli- 
cation of strength, 131 ; average amount of 
m.echanical effect produced by men and 
animals in different applications, 131 ; ex- 
ample, 132; water power, 133; example, 
133; steam, 133; example, 133; mean 
pressure at different densities, and rate of 
expansion, 134 ; pressure of steam in pounds 
per square inch — corresponding tempera- 
ture and cubic inches of steam produced by 
one inch of water, 135 ; table showing the 
weights, evaporative powers per weight, and 
bulk and character of fuels, 135 ; example, 
136. 

DEAWING OF MACHIXEET. 

Shafting, 137; described, 137; diameters of 
the journals of water-wheels and other shafts 
for heavy work, 138 ; description of cuts, 
138-139 ; the torsal strain on a shaft, 140. 
Table of diameters for shaft journals with 
reference to torsal strain, 141 ; example, 
141. Bearings or supports for the journals 
of shafts, 142; for upright shafts, 142; 
explanation of cuts, 143-144 ; pillow, 
plumber block or standard and hangers, 
145. Projections of a standard, 146 ; ex- 
planation of cuts, 147-148 ; couplings, 148 ; 
face couplings, 148 ; explanation of cuts, 
149 ; box or sleeve coupling, 150 ; horned 
coupling, 150 ; slide or clutch coupling, 
151; friction cone coupling, 152; pulleys, 
152 ; pulle3^s, cast iron, 153 ; explanation 
of cuts, 153; drums, 154; cone pulleys 
154. Table of strain on the belt by Morin, 
155 ; application of the table, 155 ; fast and 
loose pulley, 150. Geering, 157 ; spur- 
wheels, 157 ; internal geering, 158 ; rack 



VIU 



T A E L E OF CONTENTS 



geer or pinion, 158 ; bevel gcering, 158 ; 
trundle gccr, 158 ; transmission of motion, 
158 ; size of bevels, 158-159 ; the pitch of 
wheels, ICO ; rules, 161 ; example, IGl ; 
scales for geering, 1G3 ; proportion of 
scales for geering, 164 ; description of 
scales for geering, 164; application of 
scales for geering, 165-166; thickness of 
teeth, 167 ; to find the power a wheel is 
capable of transmitting, 167 ; fundamental 
principle, 168 ; epicycloidal teeth, 169 ; 
action of the teeth of a pair of wheels, 170 ; 
form of teeth, 171 ; mode of obtaining the 
curves of the teeth, 172 ; its faults, 173 ; 
involute teeth, 174. Projections of a spur 
wheel^ 175. Projections of an oMique loheel, 
178. Projections of a hevil icheel^ 179; ex- 
planation of cuts, 179-180. SlceiD devels, 181. 
System com'poseH of a 2nnion driving a racl'^ 
182; explanation of cuts, 182; system 
composed of a rack driving a pinion, 182 ; 
system composed of a wheel and tangent 
or endless screw, 183 ; system composed of 
an internal spur wheel driving a pinion, 
183 ; system composed of an internal wheel 
driven by a pinion, 184. 

Projections of Eccentrics^ 185. Definition, 
185 ; application of eccentrics, 185 ; prob- 
lems, 185; to draw the eccentrical ' sym- 
metrical curves called the heart, which 
is, &c., 185 ; a double eccentric impart- 
ing uniform motion of ascent and descent, 
186 ; other problems, 187 ; circular ec- 
centric, 188. Draicing of screws, 180 ; 
projections of a triangular-threaded screw 
and nut, 189 ; projections of a square- 
threadecl screw and nut, 190 ; doublc- 
tlireaded screw, 190 ; three-threaded screw, 
190 ; size and proportion of bolts, 190 ; 
the thread, 191 ; the nut, 191. IIools, 192. 
Frames, 192; explanation of cuts, 192; 
principle of the action of the hj-drostatic 
press, 193 ; representation of the elevation 
of the frames of three classes of American 
marine engines, 193; representation of the 
frame-work of the New World, 193 ; rep- 
resentation of the frame-work of the Paci- 
fic, 194; representation of the frame-work 
of the steamer 8ns(|uehanna, 195 ; working 
])eam of the New AVorld, 195 ; safe rule for 
land engines, 195 ; crank of Amei'ican river 
boats, 196 ; cast iron connecting rods, 197; 
wrouglit iron connecling I'ods, 197. 

Loral inn of MachiiuH, 198; examples, 199- 
200. MarhiiKH, 201 ; working of ii mncliine 
illiislrakMl. 201 ; JMuudsley i\: FieUVs direct 



action double cylinder marine engine, 201 ; 
sectional elevation through centre of the 
cylinder, of a Cornish engine, 202; longi- 
tudinal section of a locomotive boiler, 204 ; 
section through the air pump of one of the 
oscillating engines of the Golden Gate, 204 ; 
vertical section through the centre of a 
turbine wheel and the axis of supply pipe, 
205 ; rules for proportioning turbines, 207. 

AECHITECTUEAL DEAWIXG. 

Art of architecture, 209. Foundations, 210 ; 
foundations natural, 210 ; foundations ar- 
tificial, 210; foundations partly natural 
and partly artificial, 211 ; sheet piling, 211 ; 
foundations beneath the surface of water, 
212; caissons, 212; beton, 212. Walls, 
212 ; definition of terms, 213 ; walls of 
buildings, 215 ; rating of buildings in Liv- 
erpool, 215 ; building act in New York in 
regard to the thickness of walls, 216; mor- 
tar, 216 ; arches, 217 ; definition of terms, 
217 ; table of co- efficient of horizontal 
thrust at the crown, 218; examples, 218; 
dimensions of arches of European bridges, 
219 ; section of the Croton aqueduct, 219. 
Framing, 220 ; flooring, 220 ; bridging, 
220; size of joist, 221; floors, 222; fire- 
proof floors, 223 ; fire-proof floors, French, 
223 ; partitions, 223 ; frame of a wooden 
house, 224. Roofs, 224; pitch of roofs, 
225 ; explanation of cuts, 225 ; size and pro- 
portion of the different members of a roof, 
226 ; to calculate the dimensions of tie- 
rods, 226 ; example, 227 ; to find the pres- 
sure on main rafters, 228 ; example, 228 ; 
dimensions of portions of roof recommend- 
ed by Gwilt, 228. Joints, 229 ; circular 
roofs, 230 ; iron roofs, 231 ; dim.onsions 
and weights of different parts, 231 ; expla- 
nation of cuts, 232 ; general princijjles of 
bracing, 223 ; results of overloading at 
points, 234 ; on the truss b}" tension rod, 
235 ; example, 235. 

Drawing, 237. Description, 238 ; mode of 
proceeding, 238-239 ; plans of New York 
houses, 240; plans of English basement 
houses, 240 ; size and proportion of rooms 
in general, 241; dining or eating rooms, 
242 ; show room parlors, 242 ; pantries, 
243 ; passages, 243 ; lieight o{ stories, 243 ; 
details of parts, 244; stairs, 244; doors, 
245 ; explanation of cuts, 246) ; windows, 
246; l''ri'iich windows, 247; fire-])laces, 
249; roof, 250 ; eliinuiey, 250 ; nioiddings, 
251 : to i-oiistniet a liliet, 251 ; to construct 



TABLE OF C O X T E X T S . 



IX 



a torus or an astragal, 251 ; to construct an 
ovolo. 251; a Roman ovolo, 251; a Greek 
OYolo, 251; to describe a cavetto, 252; to 
describe a cyma recta or ogee, 252 ; to de- 
scribe a cjma, reversa or talon. 252 ; to de- 
scribe a scotia, 252. 

Orders of ArcMtecture^ 253. Description and 
definitions. 253; examples of parts, 254; 
example of the Ionic order, 255 ; example 
of the Corinthian order, 256 ; one of the 
capitals of the Tower of TTinds, 25G ; 
the Composite order, 257 ; the Greek or- 
ders the rudiments of modern architecture, 
259. Illustrations of the different modes 
of treatment of the arch and entablature, 
260 ; general characteristics of the Gothic, 
260 ; mouldings, 261 ; Gothic mouldings, 
261 ; jamb mouldings, 261 ; arch mould- 
ings, 262; capitals, 262; bases, 262; string 
courses, 262 ; cornices, 262 ; scroll mould- 
ing, 263. Arches. 263 ; domes and vaults, 
264 ; fan tracery yaulting, 265 ; pinnacles. 
267 ; spires, 267 ; towers, 268 ; windows, 
269 ; the lancet, 269 ; decorated traceiy, 270 ; 
doorways, 271. Italian schools, 272 ; Vene- 
tian schools, 272; Roman schools, 272. 
Ornament, 273 ; characteristic of Roman 
ornament, 274 ; characteristic of Greek and 
Roman, 274; Christian art, 275; Saracenic 
period, 275 ; Alhambra diaper, 276 ; early 
English, 276 ; ornaments of the Renais- 
sance, 277 ; ornament in the Cinquecento 
style, 278 ; Louis Quatorze style, 279 ; 
Louis Quinze style, 279 ; balusters, 279. 

Elevations of Houses^ 280. Examples, 280- 
285 ; store and warehouses, 286 ; front ele- 
vation, 287 ; school-houses, 288 ; require- 
ments of a school-house, 289; lecture 
rooms, churches, theatres, legislative 
halls, 290 ; application of acoustics, 290 ; 
space occupied by seats in general, 291; 
Romanesque church, 292 ; Byzantine 
church, 292 ; Basilican church, 293 ; win- 
dows of churches, 294 ; theatre, 295. Ta- 
ble of dimensions of a few theatres, 296 : 
construction of legislative halls, 296 ; crys- 
tal palaces, 297 ; color of country houses, 
299 ; ventilation and warming, 300 ; ven- 
tilators, 302. 

Sj)ec?f cat ions, 303. Usual forms illustrated. 
303 ; mason work, 303 ; cut stone work, 
305 ; carpenters' work, 307. 

SHADIXG AXD SHADOWS. 

direct shadows and cast shadows. 313; 
to determine the shadow cast on a vertical 



wall, 314 ; to find the shadow cast by a 
straight line upon a curved surHice, &c., 
315 ; to find the shadow cast upon a verti- 
cal plane by a given circle parallel to it, 
316; explanation of cuts, 317; to find the 
outline of the shadow cast upon both 
planes of projection by a regular hexagonal 
pyramid, 318 ; to determine the limits of 
shade on a cylinder placed vertically, &c., 
318 ; to find the line of shade in a reversed 

I cone, (tc, 319 ; explanation of cuts, 320 ; 

j to define the shadows cast upon the inte- 

I rior of a hollow cylinder in section by it- 
self, &c., 321 ; to find the outline of the 
shadow cast into the interior of a hollow 
hemisphere, 323 ; to find the line of shade 
in a sphere, &c.. 324 ; to draw the line of 
shade on the surface of a ring of circular 
section, in vertical section, elevation and 
plan, 325 ; of shadows cast on the surfaces 
of grooved pulleys, 326 ; to trace the out- 
lines of the shadows cast upon the surfaces 

J of screws, nuts, &c., 327. 

JIanijndation of Sliading and Sliadoirs. MetTt- 
ods of Tinting, 328. Surfaces in the light, 
328 ; surfaces in shade, 329 ; shading by 
flat tints, 329 ; shading by softened tints 
332 ; elaboration of sliading and sliadoics, 
333 ; paper for colored drawings, 333 ; ap- 
plication of rules to complex forms, 333 ; 
shadows appear to increase in depth as 
their distance from the spectator diminish- 
es, 335 ; examples, 835-336-337. Finished 
coloring, 339 ; how to color a circular cast- 
ing, 340 ; washing or sponging, 341 ; color- 
ing wrought iron, 343 ; coloring J^rass. 343 ; 
to promote definiteness of a colored me- 
chanical drawing, 345 ; coloring wood, brick 
and stone, 347. 

TOPOGEAPniCAL DP.AWIXG. 

Explanjjtion, 349 ; illustrations, 349-350 ; on 
the representation of hills, 351 ; German 
system of representing slopes, 353 ; draw- 
ing hills by contours, 353. Plotting, 355 ; 
scales, 355 ; for farm surveys, 355 ; for 
State surveys, 356 ; for coast survey, 356 ; 
for railroad, 356 ; for canal, 356 ; for U. S. 
engineer service, 357; variation of the nee- 



dle, 



examples of plotting, 358 ; bal- 



ancing the survey, 360-361 ; plotting curves. 
363 ; profile paper, 365 ; cross sections, 
366 ; marine surveys. 367 ; finishing the 
map or plan, 369 ; lettering, 370 ; examples, 
371; Roman, 371; Itahc. 371; Gothic, 
371; Old English Scribe Black, 372; Ger- 



TABLE OF CONTENTS. 



man, 372; Clarendon, 372; small Claren- 
don, 373 ; Grecian, 373 ; Ornamental, 373 ; 
spacing of letters, 374-375. Tinted Toi^o- 
graphical Braicing^ 378 ; marbling, 381. 
' Copy big of draioings ty Pliotograpliy^ 383. 

PEESPECTIVE DEAWING. 

\ 

J Definition, 385 ; point of view, 387 ; expla- 
nation of terms, 388 ; examples, 389-392 ; 
to determine the perspective position of 
any point in the ground plane, 392; to 
draw an octagon in parallel perspective, 
393 ; to draw a circle in parallel perspec- 
tive, 394; to draw a pyramid in parallel 
perspective, 395 ; to draw a cone in paral- 
lel perspective, 396 ; to draw a square and 
cube in angular perspective, 396 ; to draw 
a perspective projection of an octagonal pil- 
lar in angular perspective, 398 ; to construct 



a circular pillar in angular perspective, 398 ; 
to draw an octagonal pyramid in angular 
perspective, 399 ; to draw a cone in angular 
perspective, 399 ; to draw in angular per- 
spective the elevation of a building, 399 ; 
to draw in angular perspective an arched 
bridge, 400 ; to draw in parallel perspective 
the interior of a room, 401 ; to draw in per- 
spective a flight of stairs, 401 ; to find the 
reflection of objects in the water, 402 ; to 
find the perspective projection of shadows, 
402. 

ISOMETEICAL DEAWIXG. 

Definition, 405 ; explanations, 405 ; to draw 
angles to the boundary lines of an Isomet- 
rical cube, 407 ; application of this species 
of projection to curved lines, 408 ; to di- 
vide the circumference of a circle, 409. 



LIST OF PLATES 



GEOMETEICAL PEOJECTIOK 

Page 

Plate I. Projection of a regular Hexagonal Pyramid; of the same on an inclined 
base, and of a transverse section of the same, by a plane perpendicular to the 

vertical, but inclined to the horizontal plane of projection, - - - - 86 

Plate II. Projections of a Prism in an upright position, inclined to the horizontal, 

and inclined to both planes of projection, _-__--- 88 
Plate III. Construction of the Conic Sections, -------90 

Plate IV. Penetration of Cylinders at right angles to each other, - - - 92 
Plate V. Penetration of Cylinders of unequal diameters meeting at an angle ; of 

a Cone by a Sphere ; of a Cylinder by a Cylindrical Ring, - - - - 94 

Plate VI. Penetration of Cylinder, Prisms, Sphere and Cones, - - - - 98 

Plate VII. Construction of Helical Curves, and of the Spiral, - - - - 100 

Plate VIII. Development of Surfaces, .---_.-- 102 

DPvAWING OF MACHINERY. 

Plate IX. Elevations and Sections of a '\Vooden, and of an Iron Water-Wheel 

Shaft, -----..-.----- 139 

Plate X. Plan and Elevation of a Standard, ------- 146 

Plate XI. Plans, Elevations and Details of a Hanger and of a Bracket, - - 148 

Plate XII. Elevation and Section of a Spur Geer, _-_-_- 176 

Plate XIII. Projection of a Spur Geer in an obhque position, - - - - 178 

Plate XIV. Elevations and Section of a Bevel Geer, ------ 180 

Plate XV. Elevation of a Back and Pinion, and of a Worm and Endless Screw, 182 



XU LISTOFPLATES. 

Fag« 

Plate XYI. Elevations of Internal Geers^ _----_.. 134 

Plate XYII. Construction of various Eccentrics, ______ 135 

Plate XVIII. Construction of Triangular and of Square-Threaded Screvrs and 

Nuts, -------- 189 

Plate XIX. Examples of Iron Frames of Tools, ------ 192 

Plate XX. ExamjDles of American Marine Engine Frames, - - - - 194 

Plate XXI. Details of Working Beams and of Cranks, ----- 195 

Plate XXII. Details of Connecting rods, - - -- - - - - 197 

Plate XXIII. Plan of the Lower Floor of a Weaving Room, with the Position of 

the Looms, ------------- 200 

Plate XXIV. Plan of the Upper Floor of the Weaving Mill, plotted on that of 

the lower, ----------..- 200 

Plate XXV. Elevation of the Cylinders and Valve Geer of a Cornish Engine, - 202 

Plate XXVI. Details of the Cataract and Valves of a Cornish Engine. - - 202 

Plate XXVII. Longitudinal Section and End View of a Locomotive Boiler, - 204 
Plates XXVIIL, XXIX. Elevations of one of the Engines of the Golden Gate, 

from Stuart's " Naval and Mail Steamers of the United States," - - _ 204 
Plates XXX., XXXL, XXXII. Section, Plan, and Details of a Turbine Wheel, 

from Francis's " Lowell Hydraulic Experiments," ----- 206-208 



AECIIITECTUEAL DEAWING. 

Plate I. Walls of Buildings, ----- 215 

Plate II. Elevations and Details of Roofs, ------- 224 

Plate III. Elevations and Details of an Iron Roof, and of the Trusses of the 

Crystal Palace, London, - - - - -231 

Plate IV. Plans, Elevations and Section of a Small House, - - - - 238 

Plate V. Plans of Dwelling Houses, -__--_-- 240 

Plate VI. Plans, Elevations and Details of Stairs and of Doors, - - - - 244 

Plate VII. Plans, Elevations and Details of Doors and Windows, - - - 247 

Plate VIII. Examples of the Tuscan order and of Entasis, _ - - - 254 

Plate IX. Example of the Doric order, -------- 254 

Plate X. " " Ionic '•--------- 255 

Plate XL " '= Corinthian order, - - 25C) 

Plate XII. Roman Entablatures, Capitals, Bases, and Sections of Gothic Col- 
umns, and capitals of Norman and Byzantine Columns, ----- 2G0 

Plate XIII. Examples of Gothic Buttresses and Spii'cs, of Roman Campaniles, 

and Russian Towers, ----------- 2G8 

Plate XIV. Examples of Byzantine, Norman, Gothic and Saracenic Windows, - 270 

Plati: XV. Examples of Byzantine, Norman, Gothic and Saracenic Doors, - - 271 
Plate XVI. Exami)le« of Greek Ornament, and Elevations and Section of a 

r,ra<-kcl, ----___-.---- 274 

IVhATi; WII. Knriclicd Cornici- from the Temple of Jupiter Stator at Rome. Ex- 

iuii])l(s (iC lumiaii Scrolls, iind of Arab(>s(]»>o Ornament, ----- 274 

Plati", XN'III. KxamplfK of (lotliif OrnanuMil, ------- 27G 

l^LATK XIX. KxnmjdcK of JUMini.^sance Ornament, ..-.-- 278 



LIST OF PLATES. XIU 

Vaga 

Plate XX. Elevation of a City House, after designs of T. Thomas & Son, - - 280 

Plate XXI. Elevation of a House in the French style of architecture, - - 281 
Plates XXII., XXIII., XXIY. and XXY. Plans and Perspective Elevations of 

Country Residences, from '' Downing's Country Houses,*' _ . . 281-284 
Plate XXVI. Elevation in Perspective of a Tenant House after designs by John 

W. Ritch, ------------- 284 

Plate XXYII. Elevation of a Store Front, ------- 28G 

Plate XXVIII. Elevation of an Iron Store Front, by D. D. Badger & Co., - - 288 

Plate XXIX. Plan and Side Elevation of a country School House, - - _ 288 

Plate XXX. Perspective Elevation of a Church in the Romanesque style, - - 294 
Plate XXXI. Perspective Elevation of a Church in the English style of Decorated 

Gothic Architecture, ----------- 294 

Plate XXXII. A Perspective View of the Interior of the Crystal Palace, at Xew 

York, -------------- 297 

SHADING AXD SHADOWS. 

Plate I. Projection of the Shadows of Lines and Surfaces upon Vertical Planes. - 314 
Plate II. Projection of the Shadows of Sohds upon both Planes of Projection, - 318 
Plate III. Determination of the Line of Shade upon various Solids, - - - 325 
Plate IV. Determination of the Line of Shade upon Triangular and Square- 
Threaded Screws and Nuts, ---------- 327 

Plate V. Illustrations of the Process of Shading by Flat Tints, - - - - 329 

Plate VI. Illustrations of the Process of Shading bj'- Softened Tints, - - - 332 

Plate VII. Examples of Finished Shading, -_---__ 335 

Plate VIII. Examples of Finished Shading and Shadows, ----- 336 

Plate IX. Triangular and Square-Threaded Screws and Nuts in Finished Shading, 338 

Plate X. Colors of Various Materials, -------- 340 

Plate XI. Stationary Engine in Color, 343 

TOPOGEAPHIGAL DEAWIXG. 

Plate I. Conventional Signs, =.»-----__- 349 

Plate II. Designs for Meridians, - - - - - -- - - 370 

Plate III. Mechanical Method of Constructing Letters, ----- 374 

Plate IV. Examples of Titles, ---------- 374 

Plate V. Map of the Harbor and City of Xew Haven, - - _ - _ 382 

Plate VI. ]Map in India Ink only, --------- 382 

Plate VII. Map in Color, ----------- 382 

Plate VIII. Plan and Section of a Lead Mine in Color, 382 

PEKSPECTIYE DPAWIXG. 

Plate I. Projection of a Square and of a Cube in Parallel Perspective, - - 390 

Plate IL Projection of a Square and of a Cube in Angular Perspective, - - 39G 
Plate III. Projection of an Octagonal and of a Circular Pillar, of an Octagonal 

Pyramid, and of a Cone in Angular Perspective, ------ 398 

Plate IV. Elevation of a Building in Angular Perspective, ----- 4C0 



XIV LIST OF PLATES. 



Pass 



Plate Y. Elevation of an Arched Bridge in Angular Perspective, and of the inte- 
rior of a room, ----_-.-____ 4qq 

Plate YI. Perspective Projection of a Flight of Stairs, and of the Eeflection of 

Objects in the TYater, ------^-... 4Q2 

Plate YII. Perspective Projection of Shadows, - - - - - - . 402 

ISOMETEICAL DEALING. 

Plate I. Isomctrical Representation of Cubes variously cut, - - _ _ 408 

Plate II. Kepresentation of a JMitre Wheel and of a Pillow Block, - - _ 409 

Plate III. Representations of Combination of Framing in Color, - . _ 409 
Plate IY. Representation of Culvert, such as were constructed beneath the Cro- 

ton Aqueduct, _--.-_ 4Qg 



1 



A.PI>LET01SrS= 



CYCLOP. EDI A OF DRAWOG. 



GEOMETRICAL DEFmrno:NS A^D TECHNICALITIES. 

K point is mere position without magnitiicle, as tlie intersection of two 
lines, or tlie centre of a circle. 

Lines are measured by length merelj, and may be straight or curved. 
Straight lines are generally designated by letters or figures at their ex- 
tremities, as the line A B, the line 1 2. Curved lines, by 

additional intermediate letters or figures, as the curved line ABC. 

A given point or given line expresses a point or line of fixed position 
or dimension. 

Surfaces or superficies are measured by length and breadth only. Tliey 
may be plane or curved. 

Solids are measm-ed by length, breadth, and thickness. The extremi- 
ties of lines are points, the boundaries of surfaces are lines, and the boun- 
daries of solids are surfaces. 
L/^ Parallel lines are lines in the same plane 



which are equally distant from each other 



Fig. 1. 



at every part (fig. 1). 
v/ Horizontal lines are such as are parallel to the horizon, or level. 

r Vertical lines are such as are j)arallel to the position of a plumb-line 
suspended freely in a still atmosphere. 

Inclined lines occupy an intermediate between horizontal and vertical 
lines. Also two lines which converge towards each other, and if produced, 
would meet or intersect, are said to incline to each other. 
1 



GEOMETRICAL DEFINITIONS AND TECHNICALITIES. 



An omgle is the opening between two straight lines whicli meet one 

another. "When several angles are at 
one point B, any one of them is expressed 
by three letters, of which the letter that 




\ 



is at the vertex of the angle, that is, at 



the point in which the straight lines that 

contain the angle meet one another, is 

pnt between tho other two letters : Thns 

the angle which is contained by the 

straight lines, AB, CB, is named the angle ABC, or CBA ; bnt if there be 

only one angle at a point, it may be expressed by a letter placed at that 

point ; as the angle at E.' 

When a straight line standing on another straight line makes the adja- 
cent angles equal to one another, each of the angles is called a Tight angle ; 
and the straight lines are said to \iQ ^perpendicular to each other (fig. 3). 
An obtuse angle is that which is greater than a right angle (fig. 4). 



/ 



FiK. 3. 



Fig. 4 




Fig. 6. 



An acute angle is that which is less than a right angle (fig. 5). 

A triangle is a flat surface bounded by three straight lines ; when the 
three sides are equal, the triangle is equilateral ; when only two of its 
sides are equal, isosceles ^ when none equal, scaline ; when one of the 
angles is a right angle, the triangle is right angled^ and then the longest 
side, or that opposite the right angle is calle'd the hypothenuse. Tlie 
upper extremity of the triangle is called the apex^ the bottom line the 
hase^ and the two other including lines the sides. 

A Quadrilateral figure is a surface bounded by four straight lines. 




r\'A. 7. Fig. S. Fig. 9. Fig. 10. 

When llic ojtposito bides are paraUc'l, it \f^ ii2Mi'(dh'logra7n ; if its angles 



I 



OEOMETEICAL DEFINITIONS AND TECHNICALITIES. 



are riglit angles, it is a rectangle (fig. 7) ; if the sides are also equal, it is a 
square (fig. 8) ; if all the sides are equal, hut the angles not right angles, 
it is a rhoinhus (fig. 9). A trajpezium has only two of its sides parallel (fig. 
10). A diagonal is a straight line joining two opj^osite angles of a figure. 
Plane figures of more than four sides are called polygons. When the 



Fis. 11. 



Fi?. 12. 



Fig. 13. 



Fi- 14. 




Pentas^on, five sides. 



Hexagon, six sides. 



Heptagon, seven sides 



Octagon, eight sides. 




Fig. 15. 



sides are equal, tliej are regular jjyolygons / of which figs. 11-14 are ex- 
amples, annexed to which are their respective designations. 

A circle is a plane figure contained by one line, 
which is called the circumference ^ and is such that all 
straight lines, drawn from a certain point wdthin the 
figure to the circumference, are equal to one another. 
And this point is called the centre of the circle. 

Tlie term circle is very generally used for the cir- 
cumference, and will be found to be employed in this 
work with this twofold meaning. 

Any straight line drawn from the centre and terminating in the cir- 
cumference is termed a radius : if drawn through the centre, and termi- 
nated at each end by the circumference, it is termed a diameter. 

An arc of a circle is any part of the circumference. 

A sector of a circle is the space enclosed by two radii and the inter- 
cepted arc. When the radii are at right angles, the space is called a quad- 
rant or one-fourth of a circle. Half a circle is called a semicircle, 

A cliord is a straight line joining the 
extremities of an arc, as a h. The space 
cut ofii:' by the chord is termed a segment. 

A tangent to a circle or other curve 
is a straight line which touches it at only 
one point, as c d touching the circle at 
only e. 

Circles are concentric w^hen described 
from the same centres. Eccentric wlien described from difierent centres. 

Triangular or other figures with a greater number of sides are insci'ihed 




geo:metkical definitions and technicalities. 





Fig. ir. 



Fig. 18. 



m, or circumscribed hy a circle, when the vertices of all their angles are in 

the circumference (fig. 17). 



A circle is inscribed in a 
straight-sided figure, when it 
is tangent to all the sides (fig. 
18). 

All regular polygons may 
be inscribed in circles, and 
circles may be inscribed in 
polygons; hence the facility 
\vitli which polygons may be constructed. 

For the measurement of angles, the circumference of a circle is divided 

into 360 equal arcs, called degrees °, 
which are again subdivided into min- 
utes ' and seconds " ; 60 minutes to a 
degree, and 60 seconds to a minute, 
the vertex of the angle being placed 
at the centre of the circle, the angle 
is measured by the arc enclosed be- 
tween the sides. Thus the angle 
DOB is measured by the arc DB ; the 
line DPI, a line drawn from one ex- 
tremity of the arc perpendicular to 
the radius passing through the other 
extremity is called the sine of the 
angle, GD is the cosine^ HB the mrsed 
sine, AB the tangent, FE the cotangent, AC the secant, and CE the cosecant. 
An ellipse is an oval-shaped curve from any point P in which, if straight 

lines be drawn to two fixed points 




._ A- 




FF^, their sum will be always the 
same. FF' are the foci, tlie line 
passing through the foci is called 
the transverse axis, the line CD 
S perpendicular to the centre of 
tliis line the conjugate axis. 

A 2mi'abola is a curve in 
whicli any point P is equally dis- 
tant from a certain fixed point 
F ami :i straight line KK' ; tlms, 



GEOMETEICAL DEFIN^ITIONS AND TECHNICALITIES. 5 

PF is always equal to PD=. F is called tlie focus^ and the line KK' the 
directrix (fig. 21). 
K 

D — y^r 




K' 




Fig. 21. 



Fi?. 22. 



An kyj>erhola is a curve from anj point P in wliicli, if two straight lines 
be drawn to two fixed points FF^ the foci, their difiference shall always be 
the same (fig. 22). 

A cycloid is the curve described by a point P in the circumference of 




Fig. 23. 




Fig. 24, 



a circle which rolls along an extended straight line imtil it has completed 
a revolution. 

If the circle be rolled on the circumference of another circle, the curve 
then described by the point P is called an epicycloid (fig. 2-^). 

Epicycloids are external or internal, according as the rolling or gener- 
ating circle revolves on the outside or inside of the fundamental circle. 
Tlie internal epicycloid is sometimes called a hypocycloid. 



OF SOLIDS. 

K prism is a solid of which the ends are equal, similar, and parallel 
straight-sided figures, and of which the other sides are parallelograms. 
When all the sides are squares, it is called a ciihe {Q.g. 25). 

K pyramid is a sohd having a straight-sided base, and triangular sides 
terminating in one point or vertex (fig. 26). 

Prisms and pyramids arc distinguished as triangular, quadrangular, 



6 



GEOMETEICAL DEFINITIONS AND TECHNICALITIES. 



pentagonal, hexagonal, &c., according as tlie base lias three, four, five, six 
sides, &c. 






Fis. 25. 



Fig. 26. 



Fig. 27. 



A sphere or glohe (fig. 27), is a solid bounded by a uniformly curved 
surface, every point of which is equally distant from the centre, a point 
within the sphere. A line passing through the centre, and terminating 
both ways at the surface, is a diameter. 






Fig. 28. 



Fig. 29. 



Fig. 80. 



A cylinder is a round solid of uniform thickness, of which the ends are 
equal and parallel circles (fig. 28). 

A cone is a round solid, with a circle for its base, and tapering uni- 
formly to a point at the top (fig. 29). 

When a solid is cut through transversely by a plane parallel to the 
base, the part cut off is a segment, and the part remaining is afrustrum of 
the solid. The latter term is usually limited to pyramids and cones. 







Fig. 81. 



Fig. 83. 



Fig. 84. 



Tlic frf /•((//( d/'on, boinidcMl by four oquilatend triangles (fig. 30). 
The Itt.r(i/i((lron, or cube, 1)oiiihUh1 by six squares (fig. 31). 
Tlie octaJitdnni, bouucU^d ])y eight equilateral triangles (fig. 32). 
The ilo(l<('((Jn(lr(»i, boiMKh'd by twelve pentagons (fig. 33). 
Tlio icos(i/i,(/i'(>ii, Ixmudcd l)_v t\venty equihiteral triangles (fig. 34-V 
Itctriilni- solids uiMv ho circuDiscriljed by s[)lieres, and spheres may be 
iiisci-ilicd in i-oL!-ul;ir solids. 



DRAWING INSTRUMENTS. 



DEAWI:^^G INSTKUMEKTS. 

Lead jpeiicil. — ^Pencils are of various qualities, distinguislied by letter 
marks, of wliicli tlie most common in use b j draftsmen are HH and HHH. 
The pencil used for drawing straight lines should be sharpened to a chisel 
edge ; for making dots and marking points (.), the pencil should have a 
round sharp point. Pencil lines intended to be made permanent in ink, 
should be drawn quite delicately. The pencil should not be held tightly ; 
a slight hold without slackness, inclined a little to the side toward which 
the line is drawn. ^Never extend the line beyond what is necessary, and 
avoid as much as possible the use of rubber, as it roughs the paper, mak- 
ing it difficult to trace a smooth line in ink, and readier to receive and 
retain dust. 

The common ritler or straight edge. — ^Rulers should be of close-grained, 
thoroughly seasoned wood, such as mahogany, maple, pear, &c. They 
should be about \ of an inch thick, bevelled a little on one edge, and from 
1 to 2|- inches wide, according to their length. Every draftsman should 
have at least two rulers, the shortest from 9 inches to a foot long, and the 
other as long as he may require in his drawing. As the accuracy of a 
drawing depends greatly on the straightness of the lines, the bevelled edge 
of the ruler should be perfectly straight. To test this, place a sheet of 
paper on a perfectly smooth board ; insert two very fine needles in an up- 
right position through the paper into the board, distant from each other 
nearly the length of the ruler to be tested ; bring the edge of the ruler 
against these needles, and draw a line from one needle to the other ; re- 
verse the ruler, bringing the same edge on the opposite side and against 
the needles, and again draw a line. If the two lines coincide, the edge is 
straight; but if they disagree, the ruler is inaccurate. When one ruler 
has been tested, the other can be examined by placing their edges against 
the correct one, and holding them between the eye and the light. 

Triangles are made of the same kinds of wood as the ruler, and some- 
what thinner, and of various sizes. They should be right-angled, with acute 
angles of 45°, or of 60° and 30°. Tlie most convenient size for general use 
measures from 3 to 6 inches on the side. A larger size from 8 to 10 inches 



DRAWING INSTBTIMENTS. 




Fig. 35. 



long on the side is convenient for making drawings to a large scale. Cir- 
cular openings are made in the body of the triangle for the insertion of the 
end of the finger to give facility in sliding the triangle on the paper. Tri- 
angles are sometimes made as large as 15 to 18 inches on the side ; but in 
this case they are framed in three pieces of abont 1^ 
wide, leaving the centre of the triangle open. The 
value of the triangle in drawing perpendicular 
lines depends on the accuracy of the right angle. 
To test this (fig. 36), draw a line with an accurate 
ruler on paper. Place the right angle of the tri- 
angle near the centre of this line, and make one 
of the adjacent sides to coincide with the line ; now 
draw a line along the other adjacent side, Avhich, 
if the angle is strictly a riglit angle, will be per- 
pendicular to the first line. Turn the triangle on 
this perpendicular side, bringing it into 
the position ABC^ ; if now^ the sides of 
the triangle agree with the line BC and 
AB, the angle is a right angle, and the 
sides straight. The straightness of the 

— ^-r hypothenuse or longest side can be tested 

like a common ruler. 
Tlie triangle is used for tlie drawing of lines parallel or perpendicular 
to each other. Thus (fig. 37), if it were required to draw lines parallel 

and perpendicular to g d, place 
one side of the triangle so as to 
coincide accurately with the given 
line c d/ keeping the triangle in 
this position wdth the riglit hand, 
bring the edge of the ruler against 
the hypothenuse of the triangle ; if 
now the ruler be held securely by 
the left hand, the triangle may be 
slid alouii; the edj^e of the ruler, 
and any line draAvn along the upper side of the triangle will be parallel to 
tlic line r (/, and tlie linos drawn along the other side of the triangle will 
he pcr])ondi(Milai' to this same line ; in this way a rectangle may be drawn 
tlir(>uij;li tlircc '/wvw jxmils without moving the ])(>sition of the ruler. 

It is evident that for tli(> drawing of parallel lines merely, cither side 
may I>e bronglit in (M.iitacl with the ruler; but the lon^vr the side in con- 




JJ 



Fk 




Fig. OT. 



DRAWma INSTRUMENTS. 



tact, tlie more accurately may tlie parallelism be preserved in sliding the 



triangle. 

The T square is a tliin "straight edge" or ruler, a, fitted at one 
end with a stock, Z», applied transversely at right angles. Tlie stock 
being so formed as to fit and slide against one edge of the drawing board, 
the blade reaches over the surface, and presents an edge of its own at right 
angles to that of the 

board, by which par- ' ^ 

allel straight lines 
may be drawn u23on 
the paper. To suit 
a 41-inch board, the ^ ' ^^ 

blade should meas- ^jg- ss. 

ure 40 inches long clear of the stock, or one inch shorter than the board, 
to remove risk of injury by overhanging at the end ; it should be 2|- 
inches broad by /^ inch thick, as this section makes it sufficiently stiff 
laterally and vertically. The tip of the blade may be secured from 
splitting by binding it with a thin strip inserted in a saw-cut. The 
stock should be 14 inches long, to give sufficient bearing on the edge of 
the board, 2 inches broad and | inch thick, in two equal thicknesses 
glued together. With a blade and stock of these sizes, a well propor- 
tioned T square may be made, and the stock will be heavy enough to act 
as a balance to the blade, and to relieve the operation of handling the 
square. Tlie blade should be sunk flush into the upper half of the stock 
on the inside, and very exactly fitted. It should be inserted full breadth, 
as shown in the figure ; notching and dovetailing is a mistake, as it weak- 
ens the blade, and adds nothing to the security. Tlie lower half of the 
stock should be only 1| inches broad, to leave a ^-incli check or lap, by 
which the upper half rests firmly on the board and secures the blade lying 
flatly on the paper. 

For the smaller sizes of board, reduce the proportions of both the blade 
and the stock. 




Fig. 



One half of the stock, c (fig. 39), is in some cases made loose, to tiu-n 
upon a brass swivel to any angle with tlie blade a, and to be clenched by 



10. 



DRAWING mSTEUMENTS. 



a screwed mit and waslier. The loose stock is useful for drawing j)arallel 
lines obliquely to the edges of the board, such as the threads of screws, 
oblique columns, and connecting-rods of steam-engines. A square of this 
sort should be rather as an addition to the fixed square, and used only 
when the bevel edge is required, as it is not so handy as the other. 

The edges of the blade should be very slightly rounded, as the pen will 
thereby work the more freely. It is a mistake to chamfer the edges, that 
is, to plane them down to a very thin edge, as is sometimes done with the 
object of insuring a correct position of the lines ; for the edge is easily 
damaged, and the pen is liable to catch the edge, and to leave ink upon it. 
A small hole should be made in the blade near the end, by which the 
square may be hung up. 

In many drawing cases will be found the parallel ruler (fig. 40), consist- 
ing of two rulers connected 
by two bars moving on 
pivots, and so adjusted that 
the rulers, as they open, 
form the sides of a ]3aral- 
lelogram. The edge of one 
of the rulers being retained 
in a position coinciding 
with, or parallel to a given line ; the other ruler may be moved, and lines 
drawn along its edge must also be parallel to the given line. Tliis instru- 
ment is only useful in drawing small parallels, and in accuracy and con- 
venience does not compare with the triangle and ruler or T square. 




Fijr. 40. 



SWEEPS AND VAKIABLE CURVES. 

For drawing circular arcs of large radius, beyond the range of the or- 
diiuiry compasses, thin slips of wood, termed sweeps, are usefully employed, 

of which one or both edo-es 

o 

are cut to the required circle. 
For curves which are not 
circular, but variously ellip- 
tic or otherwise, "universal 
sweeps," made of thin wood, 
of variable curvature, are 
very serviceable. Tlie two 

Fl«. .12. Onor.M.rlh full M.o. OXUmplo. luiVC bcCU foUnd 

11 experience In iiicel nliiK.sl all tlie iXMjuirement^^ of ordinarv draAvin^ 




froi 




DEAWIXG IXSTRU^VIENTS. 11 

practice. AYIiatever be the nature of the curve, some portion of the uni- 
versal sweep will be fonnd to coincide with 
its commencement, and it can be continued 
throughout its extent by applying successively 

such parts of the sweep as are suitable, taking . . , 

care, however, that the continuity is not in- j,j^ ^2. or,e-fourth fuu size. 

jured by unskilful junction. 

]^o varnish of any description should be applied to any of tlie wooden 
instruments used in drawing, as the best varnish will retain dust, and soil 
the paper. Use the wood in its natural state, keeping it carefully wiped. 
Yarious other materials besides wood have been used, as steel for the blades 
of the T square and the ruler ; the objection is the liability to soil the 
paper. Glass is frequently used for the ruler and the triangle, and retains 
its correctness of edge and angle, but it is too heavy, and liable of com'se 
to fracture. 



THE CO :M PASSES OR DIVIDERS. 

The best compasses are constructed with joints of two different metals, 
as steel and brass, whereby the wear is more equal, and the motion of the 
legs uniform and steady, and not subject to sudden jerks in opening or 
shutting. This motion will occasionally requii'e some adjustment to render 
it uniformly smooth, and to move stiffer or easier at pleasm*e, but so that 
they may keep steadily any position that may be given to them. Tliis ad- 
justment is performed by the application of a turnscrew to the axis of the 
joint. In the common compasses, a simple screw forms the axis, which 
may be tui*ned with a screwdriver ; but in the best made instruments, a 
steel pin passes through the joints, having at one end a head of brass 
riveted fast u]3on it, and on the other end a similar plate or nut is screwed, 
on a diameter of which are drilled two small holes for the application of a 
key (fig. 43). The points of a well made instrument should be 
of steel so tempered, as neither to be easily bent or blunted ; not 
too fine and tapering, and yet meeting closely when the com- 
passes are shut. 

Instruction for using dividers, which are applied only to 
measure and transfer distances and dimensions, may appear Fig. 43. 
superfiuous ; but there are a few simple directions which may save the 
young draughtsman much perplexity and loss of time. It is, of course, de- 
sirable to work the compasses in such a manner that, when the dimension 



12 



DRA-VVING- INSTETJMENTS. 



is taken, it may suffer no disturbance in its transfer from the scale to the 
drawing. In order to this, the instrument is to be held by the head or 
joint, the forefinger resting on the top of the joint, and the thumb and 
second finger on either side. "When held in this way, there is no pressure 
except on the head and centre, and the dimension between the points can- 
not be altered ; but if the instrument be clumsily seized by a thumb on 
one leg, and two fingers on the other, the pressure, in the act of transfer- 
ence, must inevitably contract, in some small degree, the opening of the 
comj)asses ; and if the dimension has to be set off several times, the proba- 
bility is, that no two transfers will be exactly the same. And whilst it is all 
important to keep the dimension exact, it is also desirable to manipulate in 
such a way, when setting off the same dimension a number of times, that 
the point of position be never lost. Persons unaccustomed to the use of 
compasses, are very apt to turn them over and over in the same direction, 
when laying down a number of equal measures, and this necessitates a fre- 
quent change of the finger and thumb, which direct the movement of the 
instrument ; the consequence is, either that the fixed leg is driven deep 
into the drawing, or it loses position. ^N'ow, if the movement be alternately 
above and below the line on which the distances are being set off, the com- 
passes can be worked with great freedom and delicacy, and 
without any liability to shifting. If a straight line is drawn, and 
semicircles be described alternately above and below the line, it 
will show the path of the traversing foot. If the two movements 
are tried, the superiority of the one recommended will at once be 
discovered. The forefinger rests gently on the head ; and the 
thumb and second finger, without changing from side to side, 
direct the movement for setting off any number of times that 
may be required. 

The hair comjyasses (fig. 4i) are constructed in the same man- 
ner as the common compasses. Tlie only difference consists in a 
contrivance, whereby the lower or point half of one shank can 
be moved a very small quantity either towards or from the other 
point, so that when the compasses are opened nearly to the re- 
quired extent, by the help of the screw h the points may be set 
with great precision, which cannot be done so well by the motion 
of the joints alone. 

Compasses loith movable porrits (fig. 4;")) arc a pair of compasses 

I of wliicli llic point lialT of one of tlio U\ii,-s is movable, to admit 

^'«- '*•• of a(la|)ting singly a ])en, a pencil, or a dotting ])oint. The pen 

point is used for drawing circles or arcs witli inl<. The pencil point is a 





DEAWmG INSTRUIVIENTS. 



13 



I 




Fig. 45. 

The annexed engraving 



a tube adapted to hold a piece of lead pencil for describing circles or arcs, 

and the dotting point consists of two blades, between which revolves a 

small wheel, with numerous points round its 

circumference, resembling the rowel of a spur. 

The space between the blades being supplied 

with Indian ink, as the compasses describe a 

circle or arc, each point, as the wheel revolves, 

will pass through the ink, and transfer it to 

the paper beneath, making equidistant dots in 

the circle which the compasses describe. 

The movable points have a joint in them, 
just under that part which locks into the 
shank of the compasses, by which the part be- 
low the joint may be set perpendicular to the 
plane on which the lines are described, when 
the compasses are open. 

An additional piece, called a lengthening 
bar, is frequently applied to these compasses, 
to enable them to strike larger circles, or mea- 
sure greater extents than they otherwise could, 
represents this instrument and its appendages. 

A, the compasses, with a movable point at B ; C and D, the joints to 
set each point perpendicular to the paper ; E, the pencil point ; F, the pen 
point ; Gr, the lengthening bar. 

Bow compasses. — ^These are a small pair, either having a 
point for ink or pencil, used to describe small arcs or circles, 
which they do more conveniently than large compasses. Fig. 
46 is adapted for describing arcs of a radius intermediate between 
those described by the above-named compasses, and 
those capable of being produced by the bows repre- 
sented by fig. 47. In fig. 46, the legs can be opened a 
considerable width by the joint, whilst in fig. 47, the 
opening is limited, the two blades or legs being formed 
out of one solid piece of steel, and tempered so as to 
form a spring at the upper part ; the spring of the two 
blades is then kept in obedience by an adjusting screw 
D, by which the two points may be set to any required 
degree of minuteness, and very small circles may be 
described with precision. ^ig- ^^^ 

The pen bows (figs. 48, 49) are similar in their construction to the pen- 




14 



DRAWING INSTRUMENTS. 




Fig. 48. Fig. 49. Fig. 50. 



cil bows. Ill fig. 49 there is a second joint A, hy wliicli, wlien the insti'U- 
ment is open for use, the pen may be set perpendicular, or nearly so, to 
the paper, which is essential in the nse of the di^aw- 
ing pen. 

Similar to fig. 48 in their construction are the 
spring dividers (fig. 50), particularly useful for re- 
^ JJ^ ^ peating divisions of a small but equal extent, a 
[n| M practice that has acquired the name of stepping. 

The drawing pen (fig. 51) is used for drawing 
straight lines. It consists of two blades with steel 
points fixed to a handle ; and they are so bent, that a 
sufficient cavity is left between them for the ink, when 
the ends of the steel points meet close together, or 
nearly so. Tlie blades are set with the points more 
or less open by means of a millheaded screw, so as 
to draw lines of any required fineness or thickness. 
One of the blades is framed with a joint, so that by 
taking out the screw, the blades may be completely opened, and the points 
effectively cleaned after use. The ink is to be put between the blades by 
a common pen, and in using the pen it should be 
slightly inclined in the direction of the line to be 
drawn, and care should be taken that both points 
touch the paper ; and these observations equally apply 
to the pen points of the compasses before described. 
The drawing pen should be kept close to the ruler 
or straight edge, and in the same direction during 
tlie whole operation of drawing the line. Care 
nmst be taken in holding the straight edge firmly 
with the left hand, that it does not change its posi- 
tion. 

For drawing close parallel lines in mechanical 
and architectural drawings, or to represent canals 
or roads, a double pen (fig. 52) is frequently used, 
with an adjusting screw to set the pen to any re- 
([uircd small distance. This is usually called the 
road pen. The best pricking point is a fine needle 
held in a pair of forceps (fig. 53). It is used to mark 
the intersection of lines, or to set off divisions from 
ihe ])h)tting scale and protractor. This point may 
also ho used to prick through a drawing upon an 



R 



Fit;. 52. 



Ik. M. 



DRAWING INSTRUMENTS. 



15 



intended copy, or, the needle being reversed, the eye end forms a good 
tracing point. 

For filling np the broad lines of borders, a goose qnill is often used 
with a short nib and no slit (fig. 5-t). In drawing with this pen, incline 




Fig. 54. 

the drawing-board so that the ink will follow the pen, which prevents blots 
or the accnmnlation of too ninch ink at any one point. 

The dotting point (fig. 55) resembles a drawing pen, except that the 
23oints are not so sharp. On the back blade, as seen in the engraving, is a 
pivot, on which may be placed a dotting wheel, «, resembling the 
rowel of a spnr ; the screw h is for opening the blades to remove the Q<? 
wheel for cleaning after nse, or replacing it with one of another 
character of dot. The cap <?, at the npper end of the instrnment, is 
a box containing a variety of dotting wheels, each prodncing a dif- 
ferent shaped dot. Tliese are nsed as distinguishing marks for dif- 
ferent classes of boundaries on maps ; for instance, one kind of dot 
distinguishes county boundaries, another kind town boundaries, a 
third kind distinguishes that which is both a county and a town 
boundary, &c., &c. In using this instrument, the ink must be in- 
serted between the blades above the dotting wheel, so that, as the 
wheel revolves, the points shall pass through the ink, each carrying 
with it a drop, and marking the ]3aper as it passes. It sometimes 
happens that the wheel will revolve many times before it begins to 
deposit its ink on the drawing, thereby leaving the first 23art of the 
line altogether blank, and in attempting to go over it again, the first 
made dots are liable to get blotted. This evil may be mostly reme- 
died by placing a piece of blank paper over the drawing to the very point 
the dotted line is to commence at, then begin with drawing the wheel over 
the blank paper first, so that by the time it will have arrived at the proper 
point of commencement, the ink may be expected to flow over the points 
of the wheel, and make the dotted line perfect as required. ^ — ^^^ 

Drawing pins (fig. 56) are used to hold paper down upon a (( ® )) 
drawing or other board in any required position, and in most ^^-^^ 
cases answer better than heavy weights, which are frequently V-n—-" 
used for that purpose, as the board may be shifted from place ^Jm 



16 



DRAWING mSTRUMENTS. 



to place without moving tlie paper. Tliej consist of a brass head, with 
a steel point at right angles to its plane. A rej)resents it as seen edgewise, 
and B as seen from above. 



SCALE 



\\\\\\\\\\ \\\\\V\\\\\ \\\\\Vl 



^ 



i^ 






M 



^ 



5^- 



^^ 



^ 



i 

i 

r 



; 



m 



m 



^ ^ ^ 



I 



Ic" 



•\^ 






Fig. 57 represents the usual scale to be fonnd in the common boxes of 

drawing instrnments. It contains, on its 
two sides, simply divided scales, a diagonal 
scale and a jprotractor. Tlie simply divided 
scales consist of a series of equal divisions of 
an inch, which are numbered 1, 2, 3, &c., 
beginning from the second division on the 
left hand. 

It will be seen (figure 58) that the dif- 
ferent scales are marked 30, 35, 40, &c., and 
that the upper part of the left division in 
each is subdivided into twelve equal parts, 
and that the lower part of the same division 
is subdivided into ten equal parts. If now 
these last subdivisions or tenths be consid- 
ered as units, one mile, or one chain, or one 
foot, then each primary division will repre- 
sent ten units, ten miles, ten chains, or ten 
feet, and the scale is said to be 30, 35, 40 (ac- 
cording to the scale selected) miles, chains, 
or feet to the inch. Thus, suppose that it 
were required on a scale of 30 feet to the 
inch, to lay oif 47 feet. On the scale marked 
30, place one point of the compasses or di- 
viders at 4, and bring the other point to the 
7th lower subdivisions, counting from the 
right, and we have the distance required. 
Each of the primary divisions may be re- 
garded as unit, one foot for instance ; then 
tlie upper subdivisions are twelftlis of a foot 
or inches, and the lower subdivisions tenths 
of an inch. 

In liir. 57, the scales are marked at the 



Fig. 57. 



DRAWING rNSTRUMENTS. 



17 



left, 1 in. f , -J, J, but the divisions and subdivisions are as above. In this 
fig., the primary divisions are one inch, f , |-, and J of an inch. These scales 
are more generally used for drawings of machinery and of arcliitecture, 



60M 


■•'1 


, 1 \r\-i- 


3 


11 


.. 


vl' J.;l Ml -L- 


J. 1 


s<m . 1 


I.' . I 1 


I,' 1 


L 


,L -1 L -1 ,L 1 J. 1 


(^5 


:-ri i 


. U' - l- 


L 




8 1 I 1 L 1 ,. 


IhO 


„,,,„. 


i,- 1 !. 


1 


L 


I« i ■ ,L ! L ' ' 


35 
30 




1 L 


I 


1 


/ 1 1, i I 1 




!■''■■ 


i 1. 




1. 


L 1 i. 1 I 






m 1 18 


__ ,' 1 



Fig. 58. 

while fig. 58 are for toj^ographical drawings. Tlie application of these 
scales are similar to those already described. When the primary divisions 
are considered inches, then the drawings will be each full, f , |-, or J size, 
according to the scale adopted. 

0?i the selection of the scale. — In all working architectural and me- 
chanical drawings, use as large a scale as possible ; neither depend, even 
in that case, that the mechanics employed in the construction will measure 
correctly, but write in the dimensions as far as practicable. For architec- 
tural plans, the scale of \ an inch to the foot is one of very general use, 
and convenient for the mechanic, as the common two-foot rule carried by 
all mechanics is subdivided into ^ths, jths, and sometimes sixteenths of an 
inch, and the distances on a drawing to this scale can therefore be easily 
measured by them. Tliis fact should not be lost sight of in working draw- 
ings. Wlien the dimensions are not written, make use of such scales that 
the distances may be measured by the division of the common two-foot 
rule ; thus, in a scale of \ or \ full size, 6 inches or 3 inches represent 
one foot ; in a scale of an inch to the foot or twelfth full size, each \ an 
inch represents 6 inches, \ 3 inches ; but when \ or y^ an inch to the foot, 
or any similar scale, is adopted, it is evident that these divisions camiot 
be taken by the two-foot rule. The scale should be written on every 
drawing, or the scale itself should be dra^vn on the margin. In topographi- 
cal and geodesic drawings the latter is essential, as the scale adopted fre- 
quently has to be drawn for the specific pm-pose, and the paper itself con- 
tracts or expands with every atmospheric change, and the measurements 
will therefore not agree at all times with a detached scale ; and moreover, 
a drawing laid down from such detached scale, of wood or ivory, will not 
be uniform throughout, for on a damp day the measurements will be too 
short, and on a dry day too long. Mr. Holtzapfifel has sought to remedy 
this inconvenience by the introduction oijpajper scales ; but all kinds of paper 
do not contract and expand equally, and the error is therefore only par- 
tially corrected by his ingenious substitution of one material for another. 



18 



DRAWING mSTKUMENTS. 



Diagonal scales. — ^Tlie simply divided scales give only two denomina- 
tions, primaries and tenths, or twelfths ; but more minute subdivision is 
attained by the diagonal scale, which consists of a number of primary 
divisions, one of which is divided into tenths, and subdivided into hun- 



B 





/'//[■'/// 




1 i 


n 


^ J u ij n 










- J uj-iJ L 








f, 


- jJjlTLr 










- ■ MM tttl 








(\ 












.m jTtii: i 








8 


'JiDznjji 1 








llJlrtli. T 








M / / MM M 1 i 





Fig. 59, 



dredths by diagonal 
lines (iig. 59). This 
scale is constructed in 
the following manner : 
— ^Eleven parallel lines 
are ruled, enclosing ten 
equal spaces ; the length is set off into equal primary divisions, as DE, El, 
&c. ; the first DE is subdivided, and diagonals are then drawn from the sub- 
divisions between A and B, to those between D and E, as shown in the dia- 
gram. Hence it is evident that at every j^arallel we get an additional tenth 
of the subdivisions, or a hundredth of the primaries, and can therefore obtain 
a measurement with great exactness to three places of figures. To take a 
measm-ement of (say) 168, we place one foot of the dividers on the primary 
1, and carry it down to the ninth parallel, and then extend the other foot 
to the intersection of the diagonal, wdiicli falls from the subdivision 6, with 
the parallel that measures the eight-hundredth part (fig. 60). Tlie pri- 
maries may of course be considered as yards, feet, or inches ; and the sub- 
„ « ,. ^ , ^ divisions as tenths and 

hundredths of these 
respective denomina- 
tions. 

The diagonals may 
be applied to a scale 
wdiere only one sub- 
division is required. 
Thus, if seven lines be 
(fig. 61) ruled, enclos- 
ing six equal spaces, and the length be divided into primaries, as AB, BC, 
tfcc, the first primary AB may be subdivided into twelfths by two diagonals 
/I (; ji f running from 6, the mid- 

die of AB, to 12 and 0. 
We have here a very con- 
venient scale of feet and 

inches. From C to 6 is 

/ 2 . . 

„. ,. 1 loot (I inches : and from 

Ilf. ci. ' 





yrrr 


'jzrrrr 


\ 








trn h 








" 


n 1 r 1 










Tin r 1 


^. 








TttTi iU 


^" 








trt M Lh 


■t::.- 








L t p MM t 










MM MMi i_ 










it:Lx_ J ■ M 


/ 








/ 1 r 


\i i I i 


/ ■ 1 1 



60. 




DRAWING INSTRIIMENTS. 



19 



C on tlie several parallels to the various intersections of the diagonals, we 
obtain 1 foot and any number of inches from 1 to 12. 

Plotting scales and rulers are scales of equal parts, with the divisions 
placed on a fiducial edge, by which any length may be pricked off on to the 
paper without using the compasses, whose points, by frequent use, destroy 
the fineness of the graduation. 

On the scale (fig. 57) in common boxes of drawing instruments, die 
edge of one side is divided as a protractor, for the laying out of angles. 
Tlie instrument, when by itself, consists of a semicircle of thin metal or 




horn (fig. 62), whose circumference is divided into 180 equal parts or de- 
grees (180°). In the larger protractors each of these divisions is sub- 
divided. 

Application of the protractor. — ^To lay ofi" a given angle from a given 
point on a straight line, let the straight line (2 h of the protractor coincide 
with the given line, and the point c with the given point ; now mark on 
the paper against the division on the periphery, coinciding with the angle 
required ; remove the protractor, and draw a line through the given point 
and the mark. 

The instruments already described are those to be found in the u-sual 
cases of drawing instruments, and are sufficient for all the ordinary pur- 
pose of draughtsmen ; but there are others adapted to special purpose, or of 
careful and elaborate workmanship, which are useful where great accuracy 
and finish are required, and of some of which descriptions will be given. 

Yernier scales are preferred by some to the diagonal scale already 
described. To construct a vernier scale by which a number to three 
places may be taken, divide all the primary divisions into tenths, and 



20 DRAWING INSTEUMENTS. 

jiumber these subdivisions 1, 2, 3, from left to right. Take off now 
with tlie compasses eleven of these subdivisions, set the extent off back- 
wards from the end of the first primary division, and it will reach beyond 
the beginning of this division, or zero point, a distance equal to one of the 
subdivisions. Now divide the extent thus set off into ten equal parts, 
nuirking the divisions on the opposite side of the divided line to the strokes 
marking the primary divisions and the subdivisions, and number them 1, 
2, 3, &c., backwards from right to left. Then, since the extent of eleven 
subdivisions has been divided into ten equal parts, so that these ten parts 
exceed by one subdivision the extent of ten subdivisions, each one of these 
equal parts, or, as it may be called, one division of the vernier scale, ex- 
ceeds one of tlie subdivisions by a tenth part of a subdivision, or a hun- 
dredth part of a primary division ; thus, if the subdivision be considered 
10, then from to the first division of the vernier will be 11 ; to the sec- 
ond, 22 ; to the third, 33 ; to the fourth, 44 ; to the fifth, 55 ; and so on, 
66, 77, 88, 99. 



r 


40 




9 


7 


8 1 


5 


1 


5 


i 


5 








I • 1 _l 


111! 


1 1 1 1 1 1 1 1 l-l 1 1 t -1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 




100 


1 1 1 1 1 1 1 1 1 




1 




2 








m 


.8 


o-l 


4- 


2 


l| 



Fig. 63. 

To take off the number 253 from this scale, place one point of the di- 
viders at the third division of the vernier ; if the other point be brought to 
the primary division 2, the distance embraced by the dividers will be 233, 
and the dividers must be extended to the second subdivision of tenths to 
the right of 2. If the number were 213, then the dividers would have to 
be closed to the second subdivision of tenths to the left of 2. To take off 
the number 59 from the scale, place one point of the dividers at the ninth 
division of the vernier ; if the other point be extended to the mark, the 
dividers will embrace 99, and must therefore be closed to the fourth subdi- 
vision to the left of 0. 

These numbers, thus taken, may be 253, 25*3, 2*53 ; 213, 21-3, 213 ; 59, 
5*9, .59, according as the primary divisions are taken as hundreds, tens, or 
units. 

The construction of this scale is similar to that of the verniers of theod- 
olites and surveying instruments; but, in its application to drawing, is not 
as simple as the diagonal scales, figs. 59, Gl. 

On Honu! of the plain scales in the instrument boxes will be found divi- 
sions marked as in fig. ()4. Many of the divisions here laid down have no 
api)lic:iti<)n to drawing, according to the scope of this work; a brief ex- 
planation and application will therefore only bo given. Under doiinitions 



DRAWING INSTRUMENTS. 21 

and technicalities, the signification of the terms chords, tangents, sines, and 
secants, has been defined. Tlie chord of G0° is equal to radius, or lialf 



Rim , ^1 , M i , /,| , , 5! , , el , 7I , «! 


[Lait :.,J^^. fr'o ,JoJo,4:,J 


Cho -^io • ^'9^A.Jo..A.JgJ^JoJf> 


SJTv J^ ^^«,>,,A.A«^dJill.M', ^0 ' 50 , 


jn SreanU 


+ 




Tanl-r^H-^-O^hJ^^^^^^ry , . h , ■ ,'^^. , , , 1 ^— 




^ 




Knv)^ .... ,,,,, ,jl ,.,,,,,,, , ,4r , . ,,.^, ..,, 


^. xt^! , , i 




TT 


^1 ■' ■ ! 1 ■-^■'■' 

Lot I • , in 20 s'o , IJO 


^il,,X.A!U_ . 









Fig. &4. 

the diameter. Tlie line of chords is used to set ofi" an angle, or to measure 
an angle already laid down. 

To set off an angle. — An angle of 35° for instance : open the compasses 
to the extent of 60° on the scale of chords, setting one point at A on the 
line A B, describe with the other point an arc ; again with the compasses 
open to the extent of 35° on the scale, setting one point on B, describe an 
arc, cutting the arc B C ; through this intersection and the point A, 
draAv the line A C, and we have the angle C A B, 35°. 

To measure the angle contained by the straight lines A B and A C al- 
ready laid down. Open the compasses to the extent 
of 60° on the line of chords, as before, and with this 
radius describe the arc B C, cutting A B and A C, 
produced, if necessary, in the points B and C ; then, 
extending the compasses from B to C, place one 
point of the compasses on the beginning or zero 

point, of the line of chords, and the other point will extend to the number 
upon this line, indicating the degrees in the angle BAG. 

The lines of sines ^ secants^ tangents, and seniitangents are principally 
used for the several projections, or perspective representations, of the circles 
of the sphere, by means of which maps are constructed. 

The line of rhiunhs is a scale of the chords of the angles of deviation 
from the meridian denoted by the several points and quarter points of the 
compass, enabling the na^dgator, without computation, to lay down or 
measure a ship's com-se upon a chart. 

The line of longitudes shows the number of equatorial miles in a de~ 
gree of longitude on the parallels of latitude indicated by the degrees on 
the corresponding points of the line of chords. Examjple. — A ship in lati- 
tude 60° ^N". sailing E. Y9 miles, required the difference of longitude be- 
tween the beginning and end of her coui*se. Opposite 60 on the line of 
chords stands 30 on the line of longitudes, which is, therefore, the number 
of equatorial miles in a degree of longitude at that latitude. Hence, \\ — 
2° 38', the required difference of longitude. 



22 



DKAWING INSTEUMENTS. 



>)^ &3 feX:>J ^^^ ^ ^< 



Tlie sector (fig. Q(S) consists of two flat rulers united by a central joint, 
and opening like a pair of compasses. It carries several plain scales on its 
faces, but its most important lines are in pairs, 
running accurately to the central joint. 

Plain scales on the sector. — On tlie outer edge 
of tlie sector is usually given a decimal scale from 
1 to 100 ; and in connection with it, on one of tlie 
sides, a scale of inclies and tenths. These are 
identical with the lines on the plain scale, previ- 
ously mentioned, but the latter are more commo- 
diously placed for use. On the other side we have 
logarithmic lines of numbers, sines, and tangents. 

Sectoral cloitble scales. — Tliese are respectively 
named the Lines of Lines, Chords, Secants, Sines, 
and Tangents. Tliese scales have one line on each 
ruler, and the two lines converge accurately in the 
central joint of the sector. 

Tlie principle on which the double scales are 
constructed is, that similar triangles have their 
like sides proportional. Let the lines A B, A C, 
represent the legs of the sector, and jj ^ 

A D, A E, two equal sections from 
the centre ; then, if the points B C 
and D E be connected, the lines B C 
and D E will be parallel ; therefore, 
the triangles A B C, A D E, will be 
similar, and consequently the sides 
A B, B C, A D, D E, proportional, 
thatis, asAB:BC:: AD:DE; 
so that if A D be the half, third, or 
fourth part of A B, then D E will 
be a lialf, third, or fourth part of 
I> ; and the same holds of all the 
rest. Hence, if D E be the chord, sine, or tangent 
Ki^,. oc. of any arc, or of any number of degrees to the 

radius A D, tlien B C will be the same to the radius A 1^. Tlius at every 
ojx'uiiiir <»i' llic sector, tlic frdnsrrrsc distances 1) E and C B from one ruler 
to aiiotlicr, arc ]»n>]M>rti()nal to the lateral distances, measured on the lines 
A J>, A V. it, is to be observed, that all measures are to be taken from 
the inner lines, sinco these only run accurately to the centre. 




DKAAVIXG INSTKUMENT3. 23 

The line of lines^ marked L on each leg of the sector. — ^Tliis is a line 
of 10 primaries, each subdivided into tenths, thus making 100 divisions. 
Its use is, to divide a given line into any number of equal parts ; to give 
accurate scale measures for the construction of a drawing ; to form any 
required scale ; to divide a given line in any assigned i:)roportion ; and to 
find third, fom-tli, and middle ^proportionals to given right lines. 

To divide a given line into eight eqiial j^arts. — Take the line in the com- 
passes, and open the sector so as to apply it transversely to S and 8, then 
the transverse from 1 to 1 will be the eighth part of the line. 

To form any required scale of equal ^arts. — ^Take one inch in the com- 
passes, and oi)en the sector, till this extent becomes a transverse distance 
at the division indicating the nmuber of parts in an inch of the required 
scale. 

Example. — To adjust the sector as a sccde of one inch to four chains. 
— Make one inch the transverse distance of 4 and 4 ; then the transverse 
distances of the other corresponding divisions and subdivisions will repre- 
sent the number of chains and links indicated by these divisions : thus, the 
transverse distance from 3 to 3 will represent three chains. 

To construct a scale of feet and inches in such a onanner^ that an ex- 
tent of three inches shall represent twenty inches. — Make three inches a 
transverse distance between 10 and 10, and the transverse distance of 6 and 
6 T\dll represent 12 inches. Set off this extent, divide it into 12 equal jDarts, 
each of these divisions will represent an inch. Place the figure at the 
right, and set off again the extent of the whole twelve j^arts, from to 1, 
1 to 2, etc., to represent the feet. 

Proportion. — ^Two lines being given, to find a third proportional. 

Example. — ^Tlie given lines = 2 and 6, a third proportional required. 
Take between the compasses the lateral distance of the second term 6 on any 
convenient scale, and open the sector until this distance becomes the trans- 
vei'se distance to the first t«rm 2 ; then the transverse distance of the second 
term 6, measured upon the same scale as the former, will equal 18, the 
thu'd proportional required. 

Example. — to find a fourth proportional to the munbers 2, 6, and 10. 

Take the lateral distance of the second term 6, from any convenient 
scale of equal parts, and open the sector until that quantity, or any aliquot 
part thereof, becomes the transverse distance of the first term 2, then the 
transverse distance of the third term 10, taken from the same scale of equal 
parts, will give 30, the fourth proportional required. 

Line of Chords, marked C on each leg of the sector. Tlie double scales 
of chords upon the sector are more useful than the single line of chords de- 




24 DRAWING INSTEUMENTS. 

scribed on tlie plane scale ; for on the sector, tlie radius witli wliicli the arc 
is to be described may be of any length less than the transverse distance 
of 60 and 60 when the legs are opened as far as the instrument will admit 
of. But with the chords on the plane scale, the arc described must be 
always of the same radius. 

To protract an angle BAG, which shall contain a given number of de- 
grees, suppose 36°. 

Make the transverse distance of 60 and 60 equal to the length of the 
radius of the circle, and with that opening de- 
scribe the arc B C. 

Take the transverse distance of the given de- 
grees 36, and lay this distance on the arc from 
the point B to C. 

From the centre A of the arc, draw A C, 
A JB 

Fig. 68. A B, and these two lines will contain the angle 

required. 

To protract an angle of more than 60°, divide the required angle by 2 
or 3, and set oif as above twice or thrice the arc. 

From what has been said about the protracting of an angle to contain 
a given number of degrees, it will easily be seen how to find the degrees 
(or measure) of an angle already laid down. 

Line of Polygons. — The line of polygons is chiefly useful for the ready 
division of the circumference of a circle into any number of equal parts 
from 4 to 12 ; that is, as a ready means to inscribe regular polygons of any 
given number of sides, from 4 to 12, wdthin a given circle. To do which, 
set off the radius of the given circle (which is always equal to the side of 
an inscribed hexagon) as the transverse distance of 6 and 6 upon the line 
of polygons. Then the transverse distance of 4 and 4 will be the side of a 
square ; the transverse of 5 and 5 the side of a pentagon. 

If it be -required to form a polygon, upon a given right line set off the 
extent of the given line, as a transverse distance between the points upon 
tlie line of polygons, answering to the number of sides of which the poly- 
gon is to consist, as for a pentagon between 5 and 5, or for an octagon be- 
tween 8 and 8 ; then the transverse distance between 6 and 6 will be the 
radius of a circle, whose circumference would bo divided by the given line 
iiilo llic imiiilx'r of sides required. 

All i-cirulai- polygons, whose number of sides will exactly divide 360 
(the mmilxr ..(' degrees into which all circles are supposed to be divided) 
without a rcinaiiuler, may likewise be set of!' upon tlie circumference of a 
cii-ele hy the line of chords. Thus, take the radius of the circle between 



DRAWING INSTRUMENTS. 25 

the compasses, and open tlie sector till that extent becomes the transverse 
distance between 60 and 60 upon the line of chords ; then having divided 
360 by the required number of sides, the transverse distance between the 
numbers of the quotient will be the side of the polygon required. Tlius, 
for an octagon, take the distance between 45 and 45 ; and for a polygon 
of 36 sides, take the distance between 10 and 10, &c. 

Lines of sines ^ tangents and secants. — Given, the radius of a circle, 
required the sine and tangent of 28° 30' to that radius — 

Open the sector, so that the transverse distance of 90 and 90 on the 
sines, or of 45 and 45 on the tangents, may be equal to the given radius ; 
then will the transverse distance of 28° 30', taken from the sines, be the 
length of that sine to the given radius ; or if taken from the tangents, will 
be the length of that tangent to the given radius. 

But if the secant of 28° 30' was required — 

Make the given radius a transverse distance of and 0, at the begin- 
ning of the line of secants, and then take the transverse distance of the de- 
grees wanted, viz., 28° 30'. 

A tangent greater than 45 degrees (suppose 60) is found thus : 

Make the given radius a transverse distance to 45, and 45 at the begin- 
ning of the scale of upper tangents, and then the required degrees (60) may 
be taken from the scale. 

Given the length of the sine, tangent, or secant of any degrees, to find 
the length of the radius to that sine, tangent, or secant. 

Make the given length a transverse distance to its given degrees on its 
respective scale. Tlien 

If a sine, \ ^90 and 90 on the sines ] will be 

If a tangent tinder 45°, ( the transverse J 45 and 45 on the tangents ( the ra- 

If a tangent above 45°, T distance of j 45 and 45 on the upper tangents ( dius 

If a secant, . ) ( and on the secants ) sought. 

To find the length of a versed sine to a given number of degrees, and a 
given radius. 

Make the transverse distance of 90 and 90 on the sine equal to the 
given radius. Take the transverse distance of the complement of the sine 
of the given number of degrees. If the given number of degrees is less 
than 90, subtract the complement of the sine from the radius, the remain- 
der will be the versed sine. 

If the given number of degrees are more than 90, add the complement 
of the sine to the radius, and the sum will be the versed sine. 

To open the legs of a sector, so that the corresponding double scales oj 
lines, chords, sines, tangents, may make each a right angle. 



26 



DEAWma INSTRUMENTS. 



Oil the line of lines, make tlie lateral distance 10, a transverse distance 
between 8 on one leg and 6 on tlie other leg. 

On the line of sines, make the lateral distance 90, a transverse distance 
from 45 to 45, or from 40 to 50, or from 30 to 60, or from the sine of any 
degrees to their complement. 

On the line of tangents, make the lateral distance of 45 a transverse 
distance between 30 and 30. 

Marqiioish scales (fig. 69). — ^These scales consist of a right-angled tri- 
angle, of which the hypothennse or longest side is three times the length 
of the shortest, and a rectangular rule. Our figure, which is drawn one- 
third the actual size of the instruments from, which it is taken, repre- 
sents the triangle and a rule, as being used to draw a series of paral- 
lel lines. The rule is one foot long, and has, parallel to each of its 
edges, two scales, one placed close to the edge, and the other immediately 




within this, the outer being termed the artificial, and the inner the natm-al 
scale. Tlie divisions upon the outer scale are three times the length of 
those upon the inner scale, so as to bear the same proportion to each other 
that the longest side of the triangle bears to the shortest. In the artificial 
scales, tlie zero point is placed in the middle of the edge of the rule, and 
tlie primary divisions are numbered both ways from this point to the two 
ends of the rule, and are every one subdivided into ten equal parts, each 
of which is, consequently, three times the length of a subdivision of the 
correspoiidiiig iialiiral scale. 

TIk' ti-iaiigle has a short lino drawn perpendicular to the hypothcnuse 
ncai- llic niiddhi of it, to serve as an index or pointer ; and the longest of 
tlic ollici- two sides has a sloped ed<>;e. 

To (I nun a J'nu' jKifdlhl to a (/iveii line, at a given distance from it. — 1. 
Having a])j>licd llic given distance to the one of the natural scales which 
is lonnd 1(. nieasni-e it most eonvcMiienlly, place the triangle with its sloped 



DKAWIXG INSTRUMENTS. 



27 



edge coincident witli tlic gi\'en line, or ratlier at sucli small distance from 
it, tliat the pen or pencil passes directly over it when drawn along this 
edge. 2. Set the rule closely against the hypothenuse, making the zero 
point of the corresponding artificial scale coincide with the index upon the 
triangle. 3. Move the triangle along the rule, to the left or right accord- 
ing as the required line is to be above or below the given line, until the 
index coincides with the division or subdivision corresponding to the num- 
ber of divisions or subdivisions of the natural scale, which measures the 
given distance ; and the line drawn along the sloped edge in its new posi- 
tion will be the line required. 

The natural scale may be used advantageously in setting off the dis- 
tances in a drawing, and the corresponding artificial scale in di-a^ving 
parallels at requii'ed distances. 

The advantages of Marquois's scales are : 1st, that the sight is greatly 
assisted by the divisions on the artificial scale being so much larger than 
those of the natural scale to which the drawing is constructed ; 2d, that 
any error in the setting of the index produces an error of but 



one-third the amount in the drawing. 

If the triangle be accurately constructed, these scales may 
be advantageously used for dividing lines with accuracy and 
despatch. 

Triangular cwnpasses. — ^Fig. TO represents this instrument 
closed up. That which aj)pear3 in the fi-g. as one limb A, con- 
sists of a pair of compasses of the ordinary construction. The 
single point limb B has a compass joint at <2, by which its point 
may be opened at right angles to the plane of the pair of com- 
passes A, when the three points will form a triangle. The com- 
pass joint a is firmly attached to the centre of the compasses A, 
which, by means of a nut and screw 5, may be tui-ned round 
without moving the limbs, of which it is the centre. Tlie double 
motion thus given to the point limb B (both at right angles to, 
and parallel to the plane of the compasses A), j)artakes of the 
natm-e of a imiversal joint, and enables the three points of the 
instrument to be placed at the angular points of any shaped tri- 
angle whatever. This instrument is cliiefly useful in transfer- 
ring of points from one paj)er to another. Tlie two points of 
the compasses A being set upon such points of the di-awing as have been 
already copied, the third, B, is brought upon any other point ; then, by 
^PPtyi^g t^6 points A to the corresponding points on the copy, the point 
B will establish the other and new point on the copy. 



Fig. 70. 



28 



DRAWING INSTRUMENTS. 




Wholes and halves. — For copying and reducing drawing to half size, 
compasses called wholes and halves are used (fig. 71), in which the longer 
legs being twice the length of the shorter, when the former are opened to 
any given line, the shorter ones will be i2\ to 
oi3ened to the half of that line. By their 
means then, all the lines of a drawing may 
be reduced to one-half, or enlarged to 
double their length. These compasses are 
also useful for dividing lines by continual 
bisections. 

T\\Q pr(yportional compasses (fig. 72) are 
somewhat similar in their construction to 
wholes and halves, but of more varied ap- 
plication. The principle is the same, with 
this difference, that the screw-joint C 
passes through slides moving in the slots 
of the bars, and admits of the centre being 
Fig. 71, adjusted for various relative proportions 

between the openings A B and D E. Different sets of num- 
bers are engraved on the outer faces of the bars, and by these 
the required proportions are obtained. Tlie instrument must 
be closed for adjustment, and the nut C loosened : the slide is then moved 
in the groove, until a mark across it, named the index, coincides with the 
immber required ; which done, the nut is tightened again. 

The scales usually engraved on these compasses are named Lines, 
Circles, Planes, and Solids. 

The scale of lines is numbered from 1 to 10, and the index of the slide 
being brought to any one of these divisions, the distance D E will measure 
A B in that proportion. Thus, if the index be set to 4, D E will be con- 
tained four times in A B. 

Tlie line of circles extends from 1 to 20, and the index being set to 
(say) 10, I) E will be the tenth part of the circumference of the circle, 
whofic radius is A ]>. 

Tlic line of planes, or squares, determines the proportion of similar 
areas. Hiiis, if tlic index is placed at 3, and the side of any one square be 
talvcn l»y A \\ IVoiii a scale of ecpial parts, D E Avill be the side of another 
s(|iinn' ol' OIK -lliinl the area. And if any nnnd)er be brought to the index, 
:iii(l llic same iiuiubcr be taken by A V> from a scale of equal parts, D E 
will be the ^(|nare root of that number. And in this latter case, D E will 
also lie a mean proportional Ix'tween any two numbers, whose product is 
('(jual lo A II. 



Fiff. T2. 



DRAWING INSTRUMENTS. 



29 



Tlie line of solids expresses tlie proportion between cubes and spheres. 
Thus, if tlie index be set at 2, and the diameter of a sphere, or the side of 
a cube, be taken from a scale of equal parts by A B, tlien will D E be a 
diameter or side of a sphere or cube of half the solidity. And if the slide 
be set to (say) 8, and the same number be taken from a scale of equal 
parts, then will D E measure 2 on the same scale, or the cube root of 8. 

Beam compasses (fig. 73). — "When it is required to set off with accuracy 
distances of considerable extent, or describe arcs of over a foot radius, the 
beam compass is used. Tliis instrument consists of a beam, A A, of any 
length required, generally made of well-seasoned mahogany ; upon its 




Fig. 73. 



face is inlaid throughout its whole length a slip of holly or boxwood, a a, 
upon which are engrayed the divisions or scale, either feet and decimals or 
inches and decimals, or whateyer particular scale may be required ; but 
ordinary beam compasses are constructed with a plain beam, with no scale 
whateyer. Two brass boxes, B and C, are adapted to the beam ; the latter 
may be moved, by sliding, to any part of its length, and fixed in position 
by tightening the clamp screw E. Connected with the brass boxes are 
the two points of the instrument G and H, which may have any extent of 
opening by sliding the box C along the beam, the other box, B, being 
firmly fixed at one extremity. The object to be attained in the use of this 
instrument, is the nice adjustment of the points G II to any definite dis- 
tance apart ; this is accomplished by two vernier or reading plates 1) c, each 
fixed at the side of an opening in the brass boxes to which they are attached, 
and afford the means of minutely subdividing the principal divisions a a 
on the beam, which appear through those openings. D is a clamp screw 
for a similar purpose as the screw E, namely, to fix the box B, and j)revent 
motion in the point it carries after adjustment to position. F is a slow 
motion screw, by which the point G may be moved any very minute quan- 
tity for perfecting the setting of the instrument, after it has been set as 
nearly as possible by the hand alone. 

The method of setting the instrument for use may be understood from 
the above description of its parts, and also by the following explanation of 
the method of examining and correcting the adjustment of the vernier J, 



50 



DRA"VVmG INSTEUMKNTS. 



which will occasionally get deranged ; this verification must be done by 
means of a detached scale. Thus, suppose, for example, that onr beam 
compass is divided to feet, inches, and tenths, and subdivided by the ver- 
nier to hundredths, &c. First set the zero division of the vernier to tlie 
zero of the principal divisions on the beam, by means of the slow motion 
screw F. This must be done very nicely. Then slide the box C, with its 
point G, till the zero on the vernier c exactly coincides with any principal 
division on the beam, as twelve inches or six inches, &c., which must also 
be done very accurately ; then apply the points to a similar detached 
scale, and if the adjustment is perfect, the interval of the points G H will 
measure on it the distance to wdiicli they were set on the beam. If they 
do not by ever so small a quantity, it should be corrected by turning the 
screw F till the points exactly measure that quantity on the detached scale ; 
then, by loosening the little screws which hold the vernier h in its place, 
the position of the vernier may be gradually changed, till its zero coincides 
with the zero on the beam, and then tightening the screws again, the ad- 
justment will be complete. 

Portable or turn-in compasses (fig. 74) comprise in themselves a port- 
able case of drawing instruments, consisting of a large pair of compasses 

with movable points, which are 
also so contrived, that one forms 
in itself a small pencil bow, the 
other a pen bow ; and when the 
whole instrument is put together 
and folded up, they occupy but a 
space three inches long, and may 
be carried in the pocket without 
being an incumbrance. 

Fig. Y4: represents the instru- 
ment when all its parts are to- 
gether. Tlie principal legs of the 
instrument are F and G, movable 
as usual by a joint at A. The 
Fig. 74. lower joints, B and 0, afford the 

means of setting tlie ])(»liit limbs D and E peri)endicuhir to the paper. 

Each of the ])()int limbs may be removed from the legs F and G, and 
by means of llicir Joiiils W and C, form perfect instruments, the one a pen 
how rcj.rcs('iilc(| at 11, and tlic> other a ])on('il bow, shown at IK; the point 
lind)s oi'tliesc k'sscr instruments are all adapted to slide into the principal 
legs F and (} of the larger one, Avhich are made hollow for their reception. 




DRAWING INSTRTTMENTS. 



31 



It may easily be seen from the engraving, that by reversing either of 
the jDoints in the principal instrument, it may be sn2)plied with a pen or 
pencil as may be required, leaving the other fine or plain point E or D to 
act as a centre. 

Mr. Brunei has introduced what are called Tul)ular Compasses, in 
wdiich the upper part of the legs lengthens out like the slide of a telescope, 
thus giving greater extent of radius wdien required. The movable legs are 
double, having points at one end, and a pencil or pen at the other ; and 
they move on pivots, so that the pen or pencil can be instantly substituted 
for the points, or vice versa^ and that wdth the certainty of a perfect adjust- 
ment. Tlie design is very ingenious, and offers many conveniences, but 
the instrument is too delicate for ordinary hands. "Without extreme care 
it is soon disarranged. 

Large screw dividers (fig. 75) are used for accurately dividing lines 
into a definite number of equal parts, or for setting off equal distances. 
A is the centre about which the legs A C 
and A B open or shut. B and C are 
joints, by which the point limbs may be 
set perpendicular as usual ; the extent or 
opening between the points is regulated 
by a screw passing through a socket 
F, and terminated at the other extremity 
by a milled head E, by which the screw 
is turned round. Between this milled 
head and the nearest point limb is fixed 
what is called a micrometer head, deci- 
mally divided round its outer or cylindii- 
cal edge. One turn of the screw carries 
the micrometer head completely round ; 
therefore, wdien part of a turn only is 

given to the screw, the divisions on the head show what fraction of a tm-n 
has been given, and if it be known wdiat number of turns or threads of the 
screw are equal to one inch, the points of these compasses may be thus set 
to any small definite measure of length with the utmost precision. The 
index or zero for reading the fraction of a turn of the screw is marked on 
the point limb below B. Thus this instrument may be considered as a 
beam compass of small dimensions and minute accuracy. 

The circular protractor (fig. 76) is one of the best kind of protractors. 
It is a complete circle, A A, connected with its centre by four radii, a a a a. 
The centre is left open, and surrounded by a concentric ring or collar, J, 




32 



DRAWING mSTEUMEKTS. 



wliicli carries two radial bars, e c. To the extremity of one bar is a pinion, 
(Z, working in a tootlied rack qnite ronnd tlie outer circumference of the 
protractor. To the opposite extremity of the other bar, c, is fixed a vernier, 
which subdivides the primary divisions on the protractor to single minutes, 
and by estimation to 30 seconds. This vernier is carried round the pro- 
tractor by turning the pinion d. Upon each radial bar c c, is placed a 
branch e e^ carrying at their extremities a fine steel pricker, whose points 
are kept above the surface of the paper by springs placed under their sup- 
ports, which give way when the branches are pressed downwards, and 
allow the points to make the necessary punctures in the paper. The 
branches e e are attached to the bars c <?, with a joint which admits of their 
being folded backwards over the instrument when not in use, and for pack- 
ing in its case. Tlie centre of the instrument is represented by the inter- 
section of two lines drawn at right angles to each other on a piece of plate 
glass, which enables the person using it to place it, so that the centre, or 
intersection of the cross lines, may coincide with any given point on the 
plan. If the instrument is in correct order, a line connecting the fine 
pricking points with each other would pass through the centre of the in- 
strument, as denoted by the before-mentioned intersection of the cross 
lines upon the glass. In using this instrument, the vernier should first be 
set to zero (or the division marked 360) on the divided limb, and then 



^^^.^^^--^^^-rsr^ 




ris. 76. 



jdac'cd (III ilic j)a]KM-, ho that tlio two fine steel points may be on tlie given 
line (from wliciicc olluT and aiiguhir lines are to be drawn), and the centre 
ol'llic in>jruincnt coincides willi llic given angular point on such lino. Tliis 
done, ])rcss llic ]>n»lnict(>r gently down, which will fix it in position by 
iiieuiiH of very line points on the under side. It is now ready to lay off the 



DRAWLNO INSTRUMENTS. 



33 



given angle, or any number of angles that may be required, which is done 
by turning the pinion d till the opposite vernier reads the required angle. 
Then press downwards the branches e e, which will cause the points to 
make punctures in the paper at opposite sides of the circle ; which being 
afterwards comiected, the line will pass through the given angular point, 
if the instrument was first correctly set. In this manner, at one setting of 
the instrument, a great number of angles may be laid off from the same 
point. 

It is not essential that the centre be over the given point, when applied 
to the given line, provided the pricking points exactly fall upon the line, 
for the inclined line may be transferred to pass through the given angular 
point by a parallel ruler. 

The jpentagrajpli (fig. T7) is used for the copying of drawings either on 




the same scale, on a reduced scale, or on an enlarged scale, as may be re- 
quired. It is represented (fig. 77) as in the act of copying a plan H, upon 
3 



34: DEAWING INSTRU:MENTS. 

a reduced scale li. The pentagraph consists of four rulers, A, B, C, and D, 
made of stout brass. The two longer rulers, A and B, are connected to- 
gether by, and have a motion round a centre, shown at the upper part of 
the engraving. The two shorter rulers are, in like manner, comiected with 
each other, and with the longer rulers. The whole instrument is supported 
by small pillars resting upon ivory castors, aaa^ &c., which have a motion 
in all directions. Tlie rulers A and C have each an equal number of simi- 
lar divisions, marked j, i, &c. ; and likewise a sliding index, E and F, 
which can be fixed to any divisions on the ruler by a milled-headed clamp 
screw shown in the engraving. The sliding indexes, E and F, have each 
of them a tube adapted to slide on a pin, rising from a heavy circular 
weight called the fulcrum, which acts as a centre for the whole instrument 
to turn upon when in use, or to receive a sliding holder with a pencil or a 
tracing point, as may be required. 

To explain the method of using the instrument, the engraving repre- 
sents the instrument in the act of reducing a plan to a scale of one half the 
original. For this purpose the tracing point is fixed in a socket at G, over 
the original drawing H. The pencil is placed in a similar tube or socket at 
F, over the paper, to receive the copy ; and the fulcrum is fixed to that at E, 
the scale being one half the original. The sliding indices were firsfr clamped 
at those divisions on the rulers marked \. The instrument being thus set for 
use, if correct, the three points, E, F, and G, will be in one straight line, 
as shown by the dotted line in the figure. Tliis will invariably be the case 
at whatever division the indices may be set to. Now, if the tracing point 
G be passed delicately and steadily over every line of the plan H, a true 
copy, but of one half the scale of the original, will be marked by the pencil 
at F on the paper h beneath it. Tlie fine thread represented as passing 
from the pencil quite round the further extremity of the instrument to the 
tracer at G, is to enable the draftsman at the tracing point to raise the 
pencil from the paper, whilst he passes the tracer from one part of the ori- 
ginal to another, and prevents false lines being made on the copy. Like- 
wise, it may be noticed, that the pencil holder F is represented as sur- 
nioiiiitcd by a cnp, wliich is for the purpose of putting some small shot in, 
t(j pi-ess tlie pencil heavier upon the paper, whenever such expedient may 
be found lu'cc'ssary. 

If tlic olijcct liad l)een to enlarge the drawing to doulde its scale, then 
llic 1 nicer \\\\\A liavc been placed at F, and the pencil at G. And if a copy 
))e required, retaining the scale of the original, then the slides E and F 
must !)(>, placed at llie divisions marked 1. The fulcrum must take the 
middle station, and tlie pencil and tracer those on the exterior rules A and 
\\ of the insti-unient. 



DRAWING- INSTRUMENTS. 36 

Tlie camera lucida is sometimes used for copying and reducing topo- 
graphical drawings. A descrij^tion of the use of this instrument will be 
foimd under the head of topographical drawing. 

The drawing table and drawing hoard. — ^The usual size of the drawing 
table should be from 5 to 6 feet long, and 3 feet wide, of 1^ or 2 inch white 
pine plank well seasoned, without any knots, closely joined, glued, do welled, 
and clamped. It should be fixed on a strong firm frame and legs, and of 
such a height that the draughtsman, as he stands up, may not have to 
stoop to his work. The table is usually provided with a shallow drawer to 
hold paper or drawings. Drawing tables are made portable, by having 
two horses for their supports, and a movable drawing board for the top ; 
this board is made similar to the top of the drawing table, but of inch 
boards, and barred at the ends. Yarious woods are used for the purposes, 
but white pine is by far the cheapest and best. Drawing boards should be 
made truly rectangular, and with perfectly straight sides for the use of the 
T square. Two sizes are suflacient for common purposes, 41 x 30 inches 
to carry double elephant paper with a margin, and 31 X 24 inches for im- 
perial and smaller sizes. Boards smaller than this are too light and un- 
steady in handling. 

Small boards are occasionally made, as loose panels fitting into a frame, 
flush on the drawing surface, with buttons on the back to secure it in 
position. The panel is mostly of white pine, with a hard- wood frame. 



DRAWING PAPER. 

Drawing paper, properly so called, is made to certain standard sizes as 
follows : — 



Demy 


20 inches 


by 151 


inches. 


Medium, 


22| 




in 




Eoyal, 


24 




191 




Super Eoyal, 


. 271 




m 




Imperial, . 


30 




22 




Elephant, 


. 28 




23 




Columbier, 


35 




231 




Atlas, . 


. 34 




26 




Double Elephant, 


40 




27 




Antiquarian, . 


. 53 




31 




Emperor, . 


68 




48 





Of these, Double Elephant is the largest in common use by engineers, and 
it is the most generally useful size of sheet. Demy and Imperial are the 



36 DEAWING INSTRUMENTS. 

only other sizes worth providing for a drawing establishment. Whatman's 
white paper is the quality most usually employed for finished drawings ; 
it will bear wetting and stretching without injmy, and when so treated, 
receives color readily. For ordinary working drawings, where damp- 
stretching is dispensed with, cartridge paper, of a coarser, harder, and 
tougher quality, is preferable. It bears the use of indiarubber better, re- 
ceives ink on the original undamped surface more freely, shows a fully 
better line, and as it does not absorb very rapidly, tinting lies better and 
more evenly upon it. For delicate small-scale line-drawing, the thick 
blue paper, such as is used for ledgers, &c., imperial size, answers exceed- 
ingly well ; but it does not bear damp-stretching without injury, and 
should be merely pinned or waxed down to the board. "With good man- 
agement, there is no ground to fear the shifting of the paper. Good letter 
paper receives light drawing very well ; of course it does not bear much 
fatigue. 

Large sheets, destined for rough usage and frequent reference, should 
be mounted on linen, previously damped, with a free application of paste. 
Trdcing fajper is a preparation of tissue paper, transparent and quali- 
fied to receive ink lines and tinting without spreading. "Wlien placed 
over a drawing already executed, the drawing is distinctly visible through 
the paper, and may be copied or traced directly by the ink-instruments ; 
thus an accurate copy may be made with great expedition. Tracings may 
be folded and stowed away very conveniently ; but, for good service, they 
should be mounted on cloth, or on paper and cloth, with paste. 

Tracing paper may be prepared from thick tissue paper, by sponging 
over one surface with a mixture of one part raw linseed oil and five spirits 
of turpentine ; five gills of turpentine and one of oil will go over from 
forty to fifty sheets of paper. 

Tracing cloth is a similar preparation of linen, and is preferable for its 
toughness and durability. 

Mouth Gl'ue^ for tlie sticking of the edges of drawing paper to the 
board, is made of glue and sugar or molasses ; it melts at the temfierature 
of the mouth, and is convenient for the draughtsman. 

Drawing paper may be fixed down on the drawing board by the pins 
at tlic corners, by weights, or by gluing the edges. The first is sufiicient 
when no sliading or coloring is to be applied, and if tlie sheet is not to be 
a very long time on tlic board; and it has the advantage of preserving 
tlie pajx'r in itn natural state. For shaded or tinted drawings, the paper 
must be damped and glued at the edges, as the partial wetting of paper, 
loose or fixed at the corners merely, by the water colors distorts the surface. 



DRAWING rNSTRUMENTS. 37 

Damp-stretching is done as follows : — The edges of the paper should 
first be cut straight, and, as near as possible, at right angles with each 
other ; also the sheet should be so much larger than the intended drawing 
and its margin, as to admit of being afterwards cut from the board, leaving 
the border by which it is attached thereto bv glue or paste, as we shaU 
next explain. 

The paper must first be thoroughly and equally damped with a sponge 
and clean water, on the opposite side from that on which the drawing is to 
be made. When the paper absorbs the water, which may be seen by the 
wetted side becomino^ dim, as its surface is viewed slantwise asrainst the lis^ht, 
it is to be laid on the drawing board with the Avetted side downwards, and 
placed so that its edges may be nearly 2)arallel with those of the board ; 
otherwise, in using a T square, an inconvenience may be experienced. 
This done, lay a straight flat ruler on the paper, with its edge parallel to, 
and about half an inch from one of its edges. The ruler must now be held 
fii*m, while the said projecting half inch of paper be turned up along its 
edge ; then, a piece of solid or mouth glue, having its edge partially dis- 
solved by holding it in boiling or warm water for a few seconds, must be 
passed once or twice along the turned up edge of the paper, after which, 
by sliding the mler over the glued border, it will be again laid flat, and 
the rule being pressed down upon it, that edge of the paper will adhere to 
the board. If suflicient glue has been applied, the ruler may be removed 
directly, and the edge finally rubbed down by an ivory book-knife, or by 
the bows of a common key, by rubbing on a slip of paper placed on the 
drawing paper, so that the surface of the latter may not be soiled, which 
will then firmly cement the paper to the board. This done, another but 
adjoining edge of the paper must be acted upon in like manner, and then 
the remaining edges in succession ; we say the adjoining edges, because 
we have occasionally observed, that when the opposite and parallel edges 
have been laid down first, witliout continuing the process progressively 
round the board, a greater degree of care is required to prevent undula- 
tions in the paper as it di*ies. 

Somedmes strong paste is used instead of glue ; but as this takes a 
longer time to set, it is usual to wet the paper also on the upper sm-face to 
within an inch of the paste mark, care being taken not to mb or injure the 
smiace in the process. The wetting of the paper in either case is done for 
the purpose of expanding it ; and the edges being fixed to the board in its 
enlarged state, act as stretchers upon the paper, while it contracts in dry- 
ing, which it should be allowed to do gradually. All creases or undula- 



38 DRAWING mSTKIJMENTS. 

tions by tliis means disapjDear from the surface, and it forms a smooth 
plane to receive the drawing. 

To remove the paper after the drawing is finished, cnt off inside the 
pasted edge, and remove the edge by warm water and the knife. 

With jDanelled boards, the panel is taken out, and the frame inverted ; 
the paper, being first damped on the back with a sponge slightly charged 
with water, is applied equally over the opening to leave equal margins, 
and is pressed and secured into its seat by the panel and bars. 



MOUNTINa PAPEE AND DRAWINGS, VARNISHING, ETC. 

In mounting paper uj)on canvas, the latter should be well stretched 
upon a smooth flat surface, being damped for that purpose, and its edges 
glued down, as was recommended in stretching drawing paper. Then with 
a brush spread strong paste upon the canvas, beating it in till the grain of 
the canvas be all filled up ; for this, when dry, wdll prevent the canvas 
from shrinking when subsequently removed ; then, having cut the edges 
of the paper straight, paste one side of every sheet, and lay them upon the 
canvas sheet by sheet, overlapping each other a small quantity. If the 
drawing paper is strong, it is best to let every sheet lie ^yq or six minutes 
after the paste is put on it, for as the paste soaks in, the paper will stretch, 
and may be better spread smooth uj)on the canvas ; whereas, if it be laid 
on before the paste has moistened the paper, it will stretch afterwards and 
rise in blisters when laid upon the canvas. The paper should not be cut 
off from its extended position till thoroughly dry, which should not be 
hastened, but left in a dry room to do so gradually, if time permit ; if not, 
it may be exposed to the sun, unless in the winter season, when the help 
of a fire is necessary, provided it is not placed too near a scorching heat. 

In joining two sheets of paper together by overlapping, it is necessary, 
in order to make a neat joint, to feather edge each sheet ; this is done by 
carefully cutting witli a knife halfway through the paper near the edges, 
and on the sides which are to overlap each other ; then strip oft' a feather- 
edged sli]) from each, which, if done dexterously, will form a very neat 
and efhcieiit joint when ])nt together. 

For iiioiinting and varnisliing drawings or prints, stretcli a piece of linen 
on a frame, to wliicli give a coat of isinglass or common size, paste the back 
of drawing, wliieli leave to soak, and then lay it on the linen. Wlien dry, 
give it at least four coats of well niadt^ isinglass size, allowing it to dry be- 



DRAWING INSTRUMENTS. 39 

tween eacli coat. Take Canada balsam diluted witli the best oil of turpen- 
tine, and witli a clean brush give it a full flowing coat. 



:management of the instruments. 

In constructing preparatory pencil-drawings, it is advisable, as a rule 
of general application, to make no more lines upon the paper than are ne- 
cessary to the completion of the drawing in ink ; and also to make these 
lines just so dark as is consistent with the distinctness of the work. With 
respect to the first idea, it is of frequent application : in the case, for 
example, of the teeth of spur wheels, wliere, in many instances, all that is 
necessary to the drawing of their end view in ink are three circles, one 
of them for the pitch line, and the two others for the tops and bottoms 
of the teeth ; and again, to draw the face view of the teeth, that is, in the 
edge view of the wheel, we have only to mark oif by dividers the positions 
of the lines which compose the teeth, and draw four pencil lines for the 
two sides, and the top and bottom of the elevation. And here we may 
remark the inconvenience of that arbitrary rule, by which it is by some in- 
sisted that the pupil should lay down in pencil every line that is to be 
drawn, before finishing it in ink. It is often beneficial to ink in one part 
of a drawing, before touching other parts at all ; it j)Tevents confusion, 
makes the first part of easy reference, and allows of its being better done, 
as the surface of the paper inevitably contracts dust, and becomes other- 
wise soiled in the course of time, and therefore the sooner it is done with 
the better. 

Circles and circular arcs should, in general, be inked in before straight 
lines, as the latter may be more readily drawn to join the former, than the 
former the latter. When a number of circles are to be described from one 
centre, the smaller should be inked first, while the centre is in better con- 
dition. When a centre is required to bear some fatigue, it should be pro- 
tected with a thickness of stout card glued or pasted over it, to receive the 
compass-leg. 

Indiarubber is the ordinary medium for cleaning a drawing, and for cor- 
recting errors in the pencil. For slight work it is quite suitable ; that sub- 
stance, however, operates to destroy the surface of the paper ; and by re- 
peated application, it so rufiles the surface, and imparts an unctuosity to it, 
as to spoil it for fine drawing, especially if ink shading or coloring is to be ap- 
plied. It is much better to leave trivial errors alone,, if corrections by the 
pencil may be made alongside without confusion ; as it is, in such a case, 
time enough to clear away superfluous lines when the inking is finished.. 



40 DRAWING- INSTRUMENTS. 

For cleaning a drawing, a piece of bread two days old is preferable to 
indiariibber, as it cleans the surface well and does not injure it. When ink 
lines to any considerable extent have to be erased, a small piece of damped 
soft sponge may be rubbed over them till they disappear. As, however, 
this process is apt to discolor the paper, the sponge must be passed through 
clean water, and applied again to take up the straggling ink. For ordi- 
nary small erasures of ink lines, a sharp rounded pen-blade applied lightly 
and rapidly does well, and the surface may be smoothed down by the 
thumb-nail. In ordinary working drawings, a line may readily be taken 
out by damping it with a hair pencil, and quickly applying the indiarub- 
ber ; and to smooth the surface so roughened, a light application of the 
knife is expedient. In drawings intended to be highly finished, particular 
pains should be taken to avoid the necessity for corrections, as every thing 
of this kind detracts from the appearance. 

In using the square, the more convenient way is to draw the lines off 
the left edge with the right hand, holding the stock steadily but not very 
tightly against the edge of the board with the left hand. The convenience 
of the left edge for drawing by is obvious, as we are able to use the arms 
more freely, and we see exactly what we are doing. 

To draw lines in ink with the least amount of trouble to himself, the 
mechanical draughtsman ought to take the greater amount of trouble with 
his tools. If they be well made, and of good stuff originally, they ought 
to last through three generations of draughtsmen ; their working parts 
should be carefully preserved from injury, they should be kept well set, 
and, above all, scrupulously clean. The setting of instruments is a matter 
of some nicety, for which purpose a small oil-stone is convenient. To dress 
up the tips of the blades of the pen or of the bows, as they are usually 
worn unequally by the customary usage, they may be screwed up into con- 
tact in the first place, and passed along the stone, turning upon the point 
in a directly perpendicular plane, till they acquire an identical profile. 
Being next unscrewed and examined to ascertain the parts of unequal 
tliickness round tlie nib, the blades are laid separately upon their backs on 
\h(i stoiK^, and rubbed down at the points, till they be brought up to an 
c(lg(! (.f uniform fineness. It is well to screw them together again, and to 
l)ass llieni over the stone once or twice more, to bring up any fault ; to re- 
touch tlieni also on the outer and inner side of each blade, to remove barbs 
or frasing ; and, finally, to draw them across the palm of the hand. 

Tlio China iiil<, wliicli is coininonlv used for lino-drawino- ou<rht to be 
nibbed down in w ;iter to a cerlaln degree, avoiding the sloppy aspect of 
light lining in drawings, and making the ink just so thick as to run freely 



DRAWING INSTRUIVIENTS. 41 

from tlio pen. Tliis medium degree may be judged of after a little practice 
by the appearance of tlie ink on the pallet. Tlie best quality of ink has a 
soft feel when wetted and smoothed ; free from grit or sediment, and 
musky. Tlie rubbing of China ink in water tends to crack and break away 
the surface at the point ; this may be prevented by shifting at intervals the 
position of the stick in the hand while being rubbed, and thus rounding the 
surface. 'Nov is it advisable, for the same reason, to bear very hard, as 
the mixture is otherwise more evenly made, and the enamel of the pallet 
is less rapidly worn oif. When the ink, on being rubbed down, is likely 
to be for some time required, a considerable quantity of it should be pre- 
pared, as the water continually vaporises ; it will thus continue for a 
longer time in a condition fit for application. The pen should be levelled 
in the ink, to take up a sufficient charge ; and to induce tlie ink to enter 
the pen freely, the blades should be lightly breathed upon before immer- 
sion. After each application of ink, the outsides of the blades should be 
cleaned, to prevent any deposit of ink upon the edge of the squares. 

To keep the blades of his inkers clean, is the first duty of a draughtsman 
who is to make a good piece of work. Pieces of blotting or unsized paper 
and cotton velvet, washleather, or even the sleeve of a coat, should always 
be at hand while a drawing is being inked. When a small piece of blot- 
ting paper is folded twice so as to present a corner, it may usefully be 
passed between the blades of the pen now and then, as the ink is liable to 
deposit at the point and obstruct the passage, particularly in fine lining ; 
and for this purpose the pen must be unscrewed to admit the j^aper. But 
this process may be delayed by drawing the point of the pen over a piece 
of velvet, or even over the surface of thick blotting paper ; either method 
clears the point for a time. As soon as any obstruction takes place, the 
pen should be immediately cleaned, as the trouble thus taken will always 
improve and expedite the work. If the pen should be laid down for a short 
time with the ink in it, it should be unscrewed to keep the points apart, 
and so prevent deposit ; and when done with altogether for the occasion, 
it ought to be thoroughly cleaned at the nibs. This will preserve its edges 
and prevent rusting. 

For the designing of machinery, it is very convenient to have some 
scale of reference by which to proportion the parts ; for this purjDOse, a 
vertical and horizontal scale may be drawn on the walls of the room. 



42 GEOMETKICAL PROBLEMS. 



GEOMETEICAL PEOBLEMS. 



ON STKAIGHT LIISIES. 



It is desirable that the beginner should construct the following prob- 
lems, not copying them, but adopting some scale, which will give him the 
use of the scale, and imprint the problems on his memory. 
Problem I. — To draw a sl/raight line through given points. 
Let A and B (fig. 78) be two given points, represented by the intersec- 
tion of two lines, or pricked 

J 4 .^_:§_^ — i into the surface. Surround 

the points by small circles, 

Fig. 78. . . 

when advisable for assisting 
to define their locality, as thus O ; place the straight edge at or so near 
the points, that the point of the pen or pencil may pass thi'ough them, and 
draw the line firmly and steadily. 

Lines in drawing are divided into several classes, asfull, hroJcen, dotted, 
and hroJccn and dotted, &c. ; these again are divided into^n^, medium, and 
heavy, according to the breadth of the line (fig. 79). 

Tlie lines of a problem which are either given or are to be found, and 

Full. Broken. Dotted. Broken and dotted. 



Fig. 79. 

tlio outlines ol' Jill object thiit can be seen from the point of view in which 
it is represented should be full, and either fine, medium, or heavy, accord- 
ing to the particular effect that the draughtsman wishes to give. The por- 
tions of the outline that are hidden from view, but which are requisite to 
give a (Mniiplcte idea of tlie object, should be dotted or broken. 



GEOMETRICAL PROBLEMS. 43 

The other lines are used for conventional purposes by the draughtsman, 
as boundaries of parishes or estates, or to show a change in position of an 
object, &c., &c. 

Pkob. II. — To set off a given distance along a straight line C T>^from 
a given point A on it (fig. 80). 

Take off the given distance from the scale of equal parts with the 
dividers. Set one foot of the 

dividers on A, and bring the d Aa^ - h J? o 

other foot upon the line, and rig. so. 

mark the point B, either by 

pricking with the foot of the dividers, or by a small dot with the sharp 

point of a lead pencil. 

When the distance as at A to be set off is too small to be taken off 
from the scale with accuracy, set off any convenient distance A h greater 
than the given distance ; set off from h towards A, the length by which 
A h is greater than the given distance ; the part A a will be the required 
distance. 

To set off a number of distances on a straight line, set off their succes- 
sive sums. Tims, to set off successively the distances = 10, 15, 20, set off 
as above 10, 25, and 45, always starting from the same point. The object 
of performing the operation in this manner is to avoid carrying forward 
any inaccuracy that might be made were the respective distances set off 
separately, one after the other. If the distances to be set off are equal, it 
will be more accurate to set off a distance equal either to the whole aggre- 
gate, or such a number of them as can be contained by the compasses, and 
then dividing the line into the required parts. 

Prob. III. — To divide a giveib line into two egual jparts (fig. 81). 

Open the dividers to as near as possible half the given line, place one 
point of the dividers on the end of the line, bring the other point to the 
line, and turn on this point ; if now the point of the dividers coincide with 




Fig. SI. 



the other end of the line, we have the division required ; but should the point 
of the dividers fall within or without the end of the line, divide this deficit 
or excess by the eye into two equal parts, and contract or open the dividers 



44 GEOMETRICAL PKOBLEMS. 

to this point, and apply them again as at first ; perform the operation till 
the revolution of the compasses coincides with the given line. Thus (fig. 
81), suppose it were required to divide the line A B into two equal parts, 
and the distance A C was the first guess or opening of the dividers ; turn- 
ing on the point C, the point of the dividers that was at A falls on the 
point D beyond B, keeping the point of the dividers still on the point C, 
open them till they embrace the distance C^ J, h being at or near as can 
be judged by the eye the centre of D B ; begin again from the point A with 
the distance C^ 1) contained in the dividers, and apply the distance as at 
first, dividing the deficit or excess of the two revolutions, till the point of 
the dividers that was at A falls by revolution on B. The eye, by practice, 
becomes so accustomed to this means of division, that a plan may be re- 
duced to half scale as quickly and with as little chance of mistake as by the 
proportional compasses. 

To divide a line into any number of equal parts.—Ji the number is 

divisible by two, bisect the line, or divide it into two equal parts as above, 

and continue this as long as the number remaining is divisible by two ; but 

when the number is uneven, measure the given line on the scale, divide 

numerically the length thus found by the number of parts into which it 

is to be divided, and take on the scale as accurately as possible the quotient 

thus obtained ; apply this length successively on the line, and if the last 

distance set off does not agree with the extremity of the line ; thus if, as in 

fig. 81, when the line is to be divided into two parts, the repeated length 

exceeds the line, divide this excess by the eye into as many parts as the 

given line is to be divided, and close the dividers so as to include a 

length less by one of these parts. If the point D should fall inside 

\f,/ B, divide the deficit as before by 

/'[\ the number of parts, but open 

/ 5 \ the dividers by one of these 

I parts. 

I The above problems may be 

I constructed geometrically as fol- 

! lows : — ^To hisect or divide into 

'^ ! / two equal parts a given line A B. 

\1/ From A and B, with any radius 

,/i)\ greater than half of A B, describe 

''"■ ^^' arcs intersecting each other above 

and below tlic given line, tlic line C D connecting these intersections will 

bisect A B, and also be perpendicular to it. 



GEOMETKICAL PROBLEMS. 



45 




./ 



/ 



/ 



/\ 



Prob. IY.— To divide a given line into a given numher of egucd ^arts. 

Let A B be the distance 
to be divided, for example, 
into four equal parts ; draw 
tlie line A C, making an 
acute angle with A B ; on 
A C lay off any four equal 
distances, each as near as 
may be to j of A B, con- 
nect the last division 4 with 
B, and through the other 
points 1, 2, 3, di-aw lines 
parallel to 4: B, the inter- 
sections of these lines 5, c, d^ 
with the line A B, will divide it into four equal parts. 

Peob. Y. — To draw a jperjpendicular to a straight line^ from a j^oint 
without it. 

1st Method (fig. 84). — ^From the point A, with a sufficient radius, cut 
the given line at F and G, and from these points describe arcs cutting at E ; 
through E draw A E, which will be the perpendicular required. If there be 
no room below the line, the intersection may be taken above ; that is, be- 
tween the line and the given point. This mode is not, however, likely to 
be as exact in practice as the one given. 



y 



c 
Fig. S3. 



S, 



\ 



-¥. 







A 


\- 




v»' 






> 








' c 


B 










^ / 






/ 


\y 






,;- 


-^ ^^:iE 




'-'/ 


D 



Fig. 84. 



Fig. 85. 



'^d Method (fig. 85). — From any two points B and C, at some distance 
apart, in the given line, and with radii B A, C A, respectively, describe arcs 



46 



GEOMETEICAL PROBLEMS. 



cutting at A and D. Draw the perpendicular required, A D. This method 
is useful where the given point is opposite the end of the line, or nearly so. 

Pkob. YI. — To draw a perjpendicula/p to a straight line from a given 
point A in that line. 

1st Method (fig. 86). — ^With any radius, from the given point A, in the 
given line B C, cut the line at B and C ; with a greater radius describe arcs 
from B and 0, cutting each other at D, and draw D A the perpendicular. 



X 



\ 



B 



A\ 



/D 



B 



A 

Fig. 86. 



^ 



Fig. 87. 



2(^ Method (fig. 87).— From any centre F, above B C, describe a circle 
passing through the given point A, and cutting the given line at D ; draw 
D F, and produce it to cut the circle at E ; and draw A E the perpendicular. 
This method is useful when the point A is at or near one end ; and in prac- 
tice, it is expedient in the first place to strike out a preliminary arc, of any 
convenient radius, from the point A, as any point in that arc may be 
chosen for the centre F, with the certainty that the arc from this centre 
will pass through A, without the delay of adjusting the point of the com- 
pass to it. This expedient is of general use where an arc is to be passed 
through a given point, and particularly if the point of the pencil be round 
or misshapen, and therefore uncertain. 

2>d Method (fig. 88). — From A describe an arc E C, and from E, with the 






H 



Fig. • 



A 



5/ 



/ 



Flsr. 69. 



Jl 



Fame radius, the arc A C, cutting the other at C ; through draw a line E CD, 
and Bct off C D ocpial to C E, and through D draw A D the perpendicular 



GEOMETRICAL PROBLEMS. 



47 



required. Tliis method, like tlie previous one, is useful when the point A 
is at one end. 

4:th Method (fig. 89). — ^From the given point A, set oif a distance A E 
equal to three parts, by any scale ; and on the centres A and E, with radii 
of four and five parts respectively, describe arcs intersecting at C. Draw 
A C for the perpendicular required. Tliis method is most useful on very large 
scales, where straight edges are inapplicable, as in laying down pei-pendi- 
culars or right angles on the ground ; as in laying out the corners of houses, 
beams and girders may be set square with the sides of the houses, columns 
and the like may be set perpendicularly by the same method. The 
numbers 3, 4, 5, are, it is to be observed, taken to measm-e respectively — 
the base, the perpendicular, and the slant side of the triangle A E C. Any 
multiples of these numbers may be used with equal propriety, as 6, 8, 10, 
or 9, 12, 15, whether feet, yards, or any other measure of length. 

Peob. YII. — To draio a straight line parallel to a given U?ie, at a given 
distance apart (fig. 90). 

From the centres A, B, in the given line, 
with the given distance as radius, describe 
arcs C, D ; and draw the parallel line C D 
touching the arcs. The method of drawing 
tangents will be afte;rwards shown ; mean- 
time, in all ordinary cases, the line C D may ^^^ ^^" 
be drawn by simply applying a straight edge by the. eye. 

Peob. YIII. — To draw a jyarallel through a given point. 

1st MetJwd (fig. 91). — With a radius equal to the distance of the given 
point C from the given line A B, describe the arc D from B, taken con- 
siderably distant from C ; draw the parallel through C to touch the arc D. 



r 2) 


.--*' 


"■" " ; 




{ 


J 


I B 




A/ 



\E 



Fig. 91. 



Fig. 92. 



2d Method (fig. 92). — ^From A, the given point, describe the arc F D, 
cutting the given line at F ; from F, with the same radius, describe the 
arc E A ; and set off F D equal to E A. Draw the parallel through 
the points A, D. 



48 



GEOMETKICAL PROBLEMS. 



Peob. IX. — To construct an angle 
equal to a given angle (fig. 93). 

Thus, on the line ahio construct an an- 
gle wMcli shall be equal to the given angle 
CAB. With the dividers describe the arc 
C B ; from the point a^ with the same ra- 
dius describe c 5/ with the dividers mea- 
sure the length of the arc C B, and on <? 5 
lay off this distance ; through c draw c a^ 
and we have the required angle or open- 
ing cah^ equal to the given angle CAB. 

Prob.X. — From a point A of a given 
line D E, to draw a line making an angle 
of 60° with the given line (fig. 94). 

Take any convenient distance in the 
dividers, and from A describe the arc B C. From B, with the same dis- 
tance, describe an arc, and mark the point C where the arcs cross. Draw 




Fig. 93. 



the line A, C. 
angle of 60°. 



This line will make with the given one the required 



"^c; 




A 



B 



E 



Fig. 94. Tig- 95. 

Prob. XI. — From anoint Bon a given lineDE, to draw a line making 
an angle of 45° with it (fig. 95). 

Set off any distance B a, along D E, from B. Construct a perpendicu- 
lar to D E at a, and set off on this perpendicular a c equal to « B ; draw 
through B (? a line, which will make with D E the required angle of 45°. 

Prob. XII. — To divide a given 
angle^ asB AC (fig. 96), iiito two 
equal parts. 

From the point A, or vertex of 
the angle, with any radius describe 
an arc h c ; from h and r, the inter- 
sections of the arc with the sides of 
Fig. 00. the angle, with any radius greater 




GEOMETEICAL PEOBLEMS. 



49 



tliaii half tlie arc l <?, describe two arcs intersecting each other, as at D ; 
tlirough A and D draw a line which will bisect or divide iiito two equal 
parts the angle BAG. 

Peob. Xin. — To hisect the angle contained hetween two lines, as A B and 
CD (fig. 97), tvhe7i the intersecting point or vertex of the angle is not on the 
drawing. 

Set oif a point h at any convenient distance from A B, and through this 
point draw a parallel to 
A B ; at the same distance 
from C D draw a paral- 
lel ; extend these parallels 
till the J intersect at c; bi- 
sect the angle hcd by c «, 
which will also bisect the 
angle contained between 
the lines A B and C D. 

Peob. XIY. — Through two given points, as B and C (fig. 98), to describe 
an arc of a circle with a given radius. 

From B and C, with an opening of the dividers eqnal to the given 
radius, describe two arcs crossing at A ; from A, as a centre, with the same 
radius describe an arc which will be the one required. It is to be observed, 
that there are two points A, one above and one below the line B C, from 
which, as centres, arcs can be described with the given radius, and passing- 
through B and C. 

) 




Fior. 9T. 



//^^. 



\yi:i 





Tig. 93. 



Fig. 99. 



Peob. XY. — To find the centre of a given circle, or of an arc of a circle. 

Of a circle (fig. 99). — ^Draw the chord A B, bisect it by the pei-pendic- 
ular C D, whose extremities lie in the circumference, and bisect C D for 
the centre G of the circle. 

Of an arc, or of a circunference (fig. 100). — Select the points A, B, C, in 
the cu'cumference well apart ; with one radius describe arcs from these 



50 



GEOMETRICAL PKOBLEMS. 



three points, cutting each other ; and draw the two lines D E, F Gr, through 
their intersections : the point O, where they cut, is the centre of the circle 
or arc. 





-\M 



Fig. 100. 

Pkob. XYI. — To describe a circle j[>assing through three given points. 

Join the given points A, B, C (fig. 100), and proceed as in last problem 
to find the centre O, from which the circle may be described. 

This problem is of utility : in striking out the circular arches of bridges 
upon centering, when the s]3an and rise are given ; describing shallow 
pans, or dished covers of vessels ; or finding the diameter of a fly-wheel, or 
any other object of large diameter, when only a part of the circumference 
is accessible. 

Pkob. XYII. — To describe a circle passing through three given points^ 
when the centre is not available. 

1st Method (fig. 101). — ^From the extreme points A B, as centres, de- 
scribe arcs A II, B G. Through the third point C draw A E and B F, 
cutting the arcs. Divide A F and B E into any number of equal parts, 
and set off" a series of equal parts of the same length on the upper portions 
of the arcs beyond the points E F. Draw straight lines B L, B M, &c., to 
the divisions in A F ; and A I, AK, &c., to the divisions in E G : the suc- 
cessive intersections N, O, &c., of these lines, are points in the circle re- 
quired, between the given points A and 0, wliich may be filled in accord- 
ingly. SimiLarly, tlie remaining part of the curve B C may be described. 
jj 2d Method (fig. 102).— 

Let A, D, B, be the given 
points ; draw AB, AD, DB, 




and ef, parallel to A B. Di- 
vide D A into a number of 
equal parts, 1, 2, 3, vfcc, 



GEOMETRICAL PKOBLEMS. 



51 



and from D describe arcs tliroiigli tliese points to meet cf. Divide the arc 
A e into tlie same number of equal parts, and draw straight lines from D to the 
points of division. Tlie intersections of these lines successively with the 
arcs Ij 2, 3, &c., are points in the circle, which may be filled in as before. 

Note. — Tlie second method is not perfectly true, but sufficiently so for 
arcs less than one-fourth of a circle. 

To describe the arc mechanically with three strips of hoard forming a 
triangle. — Insert two stiff pins or nails at A and B ; place the strips as 
shown in fig. 103, 
one against the pins 
at A and B, and 
having D at their 
intersection; fasten 
the two strongly to- 
gether at this point and at the base of the triangle by the third strip ; plac- 
ing the pencil at D, and keeping the edges against A and B, moving the 
triangle to the right and left, the pencil will describe the circle. 

PuoB. XYIII. — To draw a tangent to a circle froin a given point in the 
circumference. 

1st Method. — Through the given point A (fig. 104) draw the radial line 
A C, and the perpendicular F G for the tangent required. 




Fig. 103. 





Fi- 104. 



Fig. 105. 



^d Method. — ^From A (fig. 105) set off equal segments, A B, A D ; 
join B D, and draw A E parallel to it, for the tangent. This method is 
useful when the centre is inaccessible. 

Pkob. XIX. — To draw tangents to a circle from a point without it (fig. 
106). 

Draw A C from the given point A to the centre of the circle, bisect it 
at D, from D describe an arc through C, cutting the circle at E and F. 
Draw A E, A F for the required tangents. 

To construct loithin the sides of an angle a circle tangent to these sides, 
at a given distance from the vertex. — In fig. 107, to describe a circle or arc 
tangent at a and 5, equally distant from the vertex A ; draw perpendicu- 



52 



GEOMETRICAL PEOBLEMS. 



lars to A C at a, and to A B at 5; the intersection of these will be the 
centre of the required circle. 





Fig. 107. 

In the same fig., to find the centre^ the radius heing given^ and not the 
points a and t. — Draw parallels to A C and A B at a distance equal to the 
given radius, and their intersection will be the centre required. 

Pkob. XX. — To describe a circle from a given jpoint to touch a given 
circle (figs. 108, 109). 

D E being the given circle, and B the point, draw from B to the centre C, 
and produce it, if necessary, to cut the circle at A, and with B A as radius 
describe the circle F G, touching the given circle. The operation is the 
same whether the point B be within or without the circle. 




Fig. 108, 




It will be remarked that, in all cases of circles tangential to each other, 
their centres and their j)oints of contact must lie in the same straight line. 

Peob. XXI.— 
To draw tangents 
to two given circles. 
1st Jfethod (fig. 
110).— Draw the 
straight line ABC 
through the cen- 
iig. no. tres of the two 




GEOMETRICAL PROBLEMS. 



given circles ; from tlie centres A, B, draw parallel radii A D, B E, in 
the same direction ; join D E, and produce it to meet the centre line at C, 
and from C draw tangents to one of the circles by Prob. XIX. Those tan- 
gents will tonch hoth circles, as required. 

^Id 2fethod (fig. 111). — Draw A B, and in the larger circle draw any 
radius A PI, on which set off H G equal to the radius of the smaller circle ; 




Fig. 111. 

on A describe a circle with the radius A G, and cb*aw tangents B I, B K, to 
this circle from the other centre B ; from A and B draw perpendiculars to 
these tangents, and join C, D, and E, F, for the required tangents. 

JS^ote. — ^The second method is useful when the diameters of the circles 
are nearly equal. 

Prob. XXII. — Between tico inclined lines to draio a series of circles 
touching these lines and touching each other i^g. 112). 

Bisect the inclination of the given lines A B, C D, by the line ^N O ; 
this is the centre line of the circles to be inscribed. From a point P in this 
line draw the perpendicular P B 
to the line A B, and from P de- 
scribe the cii'cle B D touching the 
ffiven lines and cuttino- the centre 

c5 O 




Fig. 112. 



line at E ; from E draw E F per- 
pendicular to the centre line, cut- 
ting A B at F, and from F describe 
an arc E G, cutting A B at G ; 
draw G H parallel to B P, giving H the centre of the second touching 
circle, described with the radius H E or H G. By a similar process the 
third circle I ]N" is determined. And so on. 

Inversely, the largest circle may be described first, and the smaller 
ones in succession. 

I^ote. — This problem is of frequent use in scroll work. 

Prob. XXm. — Between tioo inclined lines to draw a circiolar segment 
to fill tvp the angle^ and touching the lines (fig. 113). 



54: 



GEOMETRICAL PKOBLEMS. 



Let A B, D E, be the inclined lines ; bisect the inclination by the line 
F Cj and draw the perpendicular A F D to define the limit within which 
the circle is to be drawn. Bisect the angles A and D by lines cutting at 
C, and from C with radius C F, draw the arc H F G as required. 





Fig. 114. 



Peob. XXIY. — To -fill uj[> the angle of a straight line and a circle^ with 
a circular arc of a given radius (fig. 114). 

In the given circle A D draw a radius B and produce it, set off B E 
equal to the radius of the required arc, and on the centre C with the radius 
C E, describe the arc E F. Draw G F parallel to the given line H I, at 
the distance G H equal to the radius of the required arc, and cutting the 
arc E F at F. Then F is the required centre ; draw the perpendicular F I, 
and the radius F C cutting the circle at A, and with the radius F A or F I 
describe the arc A I as required. 

Peob. XXY. — To fill up the angle of a straight line and a circle, with 
a circular arc to join the circle at a given jpoint (fig. 115). 

In the given circle draw the radius A and produce it ; at A draw a 

tangent meeting the given line at 
D ; bisect the angle A D E so formed . 
with a line cutting the radius at F ; 
and on the centre F describe the 
arc A E as required. 

Pkob. XX YL — To descnhe a 
circular arc joining two circles, and 
to touch one of them at a given 
2)oint (fig. IIC). 

Let A B and F G be tlic given 
circles, to bo joined by an arc 
niiii; fmc ot tiioni at I. Draw the radius E F, and produce it both 
ways ; net oil' I"' II c([\ui\ to the radius A C of the other circle ; join C H 




toiK'l 



one of thoni at F. 



GEOMETRICAL PROBLEMS. 



65 



and bisect it with the perpendicular L I cutting E F at I ; on the centre I 
J^\ with radius I F describe the arc F as required. 




a, 

Fis:. 117. 



Prob. XXyn. — To find the arc which shall 

he tangent to a given jpoint Kona straight line^ 

;-:k'_ andjpass through a given jpoint outside the line 

Fig. 116. (fig. iiY). 

Erect at A a perpendicular to the given line ; connect C A, and bisect 
it by a perpendicular ; the intersection of the two perpendiculars at a will 
be the centre of the required arc. 

Prob. XXYIII. — To connect two jparallel lines hy a reversed curve conh- 
posed of two arcs of equal radiiis^ and tangent to the lines at given points^ 
as at A and B (fig. 118). 

Join A B, and divide it into two equal parts at C ; bisect C A and C B 
bj perpendiculars ; at A and B erect perpendiculars to the given lines, and 
the intersections a and h will be the centres of the arcs composing the re- 
quired curve. 




Fi?. lis. 



Fig. 119. 



Prob. XXIX. — To join two given points^ as A and B (fig. 119), in two 
given parallel lines ly a reversed cuicve of two equal arcs, whose centres lie 
in the parallels. 

Join A B, and divide it in equal parts at C, as above. Bisect also A C 
and B C by perpendiculars ; the intersections a and 5 of the parallel lines 
by these perpendiculars will be the centre of the required arcs. 

Prob. XXX. — On a given line, as A B (fig. 120), to construct a corn- 
pound cw^e of three arcs of circles, the radii of the two side ones heing 



56 



GEOMETRICAL PEOBLEMS. 



equal and of a given lengthy and their centres in the given line / the central 
arc to pass through a given point, as C, on the perpendicular bisecting the 
given line, and tangent to the other two arcs. 

Draw tlie perpendicular C D ; lay off A <^, B J, and C c, each equal to 

tlie given radius of the 
side arcs ; join a c; 
bisect a c \)j 2i per- 
pendicular ; the inter- 
section of this line 
with the perpendicu- 
lar C D will be the 
required centre of the 
central arc. Through 
a and h di*aw the lines 
D e and D e' ; from a 
and h with the given radius, equal to a A, h B, describe the arcs A e and 
B e\ from D as a centre, with a radius equal to C D, and consequently by 
construction D e and D e' , describe the arc e C e' , and we have the com- 
pound curve required. 

For the construction of compound curves of five arcs, see construction 
of ellipses, jDage 72. 




PROBLEMS ON CIRCLES AND RECTILINEAR FIGURES. 

Prob. XXXT. — To construct a triangle upon a given straight line or 
hase, the length of the two sides heing given. 

First, an equilateral triangle (fig. 121). On the ends A B of the given 
line, with A B as radius, describe arcs cutting at C, and draw A C, B C ; 
then A B C is the triangle required. 

•■■.€..-■■ 



A 



\ 

B 




Fls. 122. 



Flp. 121. 

Second, wlicii flic nidcs arc unequal (fig. 122). Let A D bo the base, 
and 1) and C tli(> two si<U's. On citlier end, as A, of the base-line, with the 



GEOMETRICAL PROBLEMS. 



57 



line B as radius, describe an arc ; and on D, with C as radius, cut the arc 
at E. Draw A E, E D, then A E D is the triangle as required. 

This construction is used also to find the position of a point, when its 
distances are given from two other given points, whether joined by a line 
or not. 

Peob. XXXII. — To construct a square or a rectangle upon a given 
straight line. 

Eirst, a square (fig. 123). Let A 33 be the given line ; on A and B as 
centres, with the radius A B, describe arcs 
cutting at C ; on C, with the same radius, 
describe arcs cuttincr the others at D and 



E ; and on D and E, cut these at F G. 




>? 



Draw A F, B G, cutting the arcs at H, I ; 
and join H I to form the square as re- 
quired. 

Second, a square or rectangle (fig. 124). 
To the base E F draw perpendiculars E H, 
F G, equal to the sides, and join G H to complete the rectangle. 

When the centre lines of the square or rectangle are given, the figure 
may be described as follows : — 



Fis. 123. 








c. 






K 


If 


L 




'■' 


'^ ! ~"~ 


-" 


""'" 


^,1 


F___ 


t:} 


G 


M 


. 




i ,. 


. 


»--' 


I 


i 


J; 




IVf 




m 








Fig. 125. 







Fig. 124. 

Let A B and C D (fig. 125) be the centre lines, peiqDendicular to each 
other, and E the middle point of the figure ; set oflP E F, E G, equal each 
to the half length of the rectangle, and E H, E J, each equal to half the 
height. On the centres H, J, ^^dth a radius equal to the half length, de- 
scribe arcs on both sides ; and on F, G, with a radius of half the height, 
cut these arcs at K, L, M, j^. Join the four intersections so formed, to 
complete the rectangle. 

Pkob. XXXIII. — To construct a jparallelogram^ of which the sides and 
one of the angles are given (fig. 126). 

Let A and B be the lengths of the two sides, and C the angle ; draw a 
straight line, and set ofl^ D E equal to A ; from D di*aw D F equal to B, 
and forming an angle with D E equal to C ; from E with D F as radius, 



58 



GEOMETRICAL PROBLEMS. 



describe an arc, and from F ^itli D E as radius, cnt the arc at G, and 
draw F G and E G, to complete the parallelogram. Or, the remaining 
sides may be drawn parallel to D E and D F, cutting at G, and the figure 
is thus completed. 

c 



R 





-A- 



-B- 




Fig. 126. 



Fig. 12T. 



Peob. XXXIY. — To describe a circle about a triangle (fig. 127). 

Bisect two of the sides A B, A 0, of the triangle at E, F ; from these 
points draw perpendiculars cutting at K. From the centre K, w4th K A 
as radius, describe the circle A B C, as required. 

Peob. XXXV. — To inscribe a circle in a triangle (fig. 128). 

Bisect two of the angles A, C, of the triangle ABC, by lines cutting 
at D ; from D draw a perpendicular D E to any side, as A C ; and with 
D E as radius, from the centre D, describe the circle required. 

Wlien the triangle is equilateral, the centre of the circle is more easily 
found by bisecting two of the sides, and drawing perpendiculars, as in the 
previous problem. Or, draw a perpendicular from one of the angles to 
the opposite side, and from the side set off one-third of the length of the 
peq)endicular. 





Fig. 120. 



Prob. XXXYI. — To inscribe a square in a circle; and to describe a 
circle about a square (fig. 129). 



GEOMETRICAL PROBLEMS. 



59 



To inscribe the square. Draw two diameters A B, C D, at right angles, 
and join the points A, B, C, D, to form the square as required. 

To describe the circle. Draw the diagonals A B, C D, of the given 
square, cutting at E ; on E as a centre, with E A as radius, describe the 
circle as required. 

In the same way, a circle may be described about a rectangle. 

Prob. XXXYII. — To inscribe a circle in a square; and to describe a 
square about a circle (fig. 130). 

To inscribe the circle. Draw the diagonals A B, CD, of the given 
square, cutting at E ; draw the perpendicular E F to one of the sides, and 
with the radius E F, on the centre E, describe the circle. 

To describe the square. Draw two diameters A B, C D, at right angles, 
and produce them ; bisect the angle D E B at the centre by the diameter 
F G, and through F and G draw perpendiculars A C, B D, and join the 
points A, D, and B, C, where they cut the diagonals, to complete the square. 

Prob. XXXYIII. — To inscribe a pentagon in a circle (fig. 131). 

Draw two diameters A C, B D, at right angles ; bisect A O at E, and 




Fig. 182. 



from E with radius E B, cut A C at F ; from B, with radius B F, cut the 
circumference at G H, and with the same radius step round the circle to I 
and K ; join the points so found to form the pentagon. 

Prob. XXXIX. — To construct a regular hexagon u])on a given straight 
line (fig 132). 

From A and B, wdtli a radius equal to the given line, describe arcs 
cutting at g ; from ^, with the radius g A, describe a circle ; with the same 
radius set off from A the arcs A G, G F, and from B the arcs B D, D E. 
Join the points so found to form the hexagon. 

Prob. XL. — To inscribe a regidar hexagon in a circle (fig. 133). 

Draw a diameter A B, from A and B as centres, with the radius of the 
circle A C, cut the circumference at D, E, F, G ; draw straight lines A D, 



D E, &c., to form the hexagon. 



60 



GEOMETKICAL PKOBLEMS. 



The points of contact, D, E, &c., may also be found by setting off tbe 
radius six times npon the circumference. 





-' G - 



Fig. 133. 



Fisr. 134. 



Peob. XLI. — To describe a regular hexagon about a circle (fig. 134). 

Draw a diameter A B of the given circle ; with the radius A D from A 
as a centre, cut the circumference at C ; join A C, and bisect it with the 
radius D E ; through E draw E G parallel to A C, cutting the diameter at 
F, and with the radius D F describe the circle F H. Within this circle 
describe a regular hexagon by the preceding problem ; the figure will touch 
the given circle as required. 

Peob. XLII. — To construct a regular octagon upon a given straight line 
(fig. 135). 

Produce the given line A B both ways, and draw pei^Dendiculars A E, 
B F ; bisect the external angles at A and B by the lines A H, B C, which 
make equal to A B ; draw C D and H G parallel to A E and equal to A B ; 
and from the centres G, D, with the radius A B, cut the perpendiculars at 
E,F, and draw E F to complete the octagon. 



Bk 



A B 

Fiz. m'). 




Fig. 18G. 



Pkc.u. 

Draw 
with A e 
]t< Milts so 



XLTTT. — To convert a square into a regular octagon (fig. 130). 
tlic (liai^oiials of the scpiare cutting at ^y fn^iu the corners A,P), C,D, 
as radius, descnl)e ai'cs cutting tlie sides at [/ //, c^^c. ; join the 
found to (•oin])U'te the octagon. 



GEOMETRICAL TEOBLEMS. 



61 



Pkob. XLIY. — To inscribe a regular octagon in a circle (fig. 137). 
Draw two diameters A C, B D, at right angles ; bisect tlie arcs A B, 
B C, &c., at ^,/, &c. ; and join A ^, <? B, <fec., for the inscribed figure. 





Fig. 138. 

Pkob. XLY. — To describe a regular octagon about a circle (fig. 138). 

Describe a square about the given circle A B ; draw perpendiculars 
h A', &c., to the diagonals, touching the circle. The octagon so formed is 
the figure required. 

Or, to find the points A, ^, &c., cut the sides from the corners of the 
square, as in Prob. XLIII. 

Prob. XLYI. — To inscribe a circle within a regular polygon. 

When the polygon has an even number of sides, as in fig. 139, bisect 
two opposite sides at A and B, draw A B, and bisect it at C by a diagonal 
D E drawn between opposite angles ; with the radius C A describe the 
circle as required. 

When the number of sides is odd, as in fig. 140, bisect two of the sides 
at A and B, and draw lines A E, B D, to the opposite angles, intersecting 
at C ; from C with C A as radius, describe the circle as required. 





Fis. 140. 



Prob. XLYII. — To describe a circle without a regidar jpolygon. 
When the number of sides is even, draw two diagonals from opposite 

angles, like D E (fig. 139), to intersect at C ; and from C with C D as 

radius, describe the circle required. 



6^ 



GEOMETRICAL PROBLEMS. 



Wlien the number of sides is odd, find the centre C (fig. 140) as in last 
problem, and with C D as radius describe the circle. 

The foregoing selection of problems on regular figures is the most use- 
ful in mechanical practice on that subject. Several other regular figures 
may be constructed from them by bisection of the arcs of the circumscrib- 
ing circles. In this way a decagon, or ten-sided polygon, may be formed 
from the pentagon by the bisection of the arcs in Prob. XXXYIII., fig. 
131. Inversely, an equilateral triangle may be inscribed by joining the 
alternate points of division found for a hexagon. 

The constructions for inscribing regular polygons in circles are suitable 
also for dividing the circumference of a circle into a number of equal 
parts. To su23ply a means of dividing the circumference into any number 
of parts, including cases not provided for in tlie foregoing problems, the 
annexed table of angles relating to polygons, expressed in degrees, will be 
found of general utility. In this table, the angle at the centre is found by 

Table of Polygon^al Angles. 



Number of Sides of Regular 








Polygon; or number of 


Angle at 


Number of Sides of Regular 


Angle at 


equal parts of the circum- 
ference. 


Centre. 


Polygon. 


Centre. 


No. 


Degrees. 


No. 


Degrees. 


3 


120 


12 


30 


4 


90 


13 


27/3 


5 


72 


14 


25f 


6 


60 


15 


24 


r 


51| 


16 


224- 


8 


45 


17 


21t\ 


9 


40 


18 


20 


10 


36 


19 


1811 


11 


32A 


20 


18 



dividing 360°, the number of degrees in a circle, by the number of sides 
in Ibe polygon ; and by setting off round tlie centre of the circle a suc- 
cession of angles by means of the protractor, equal to tlie angle in the 
tabic due to a given number of sides : the radii so drawn will divide the 
circumference into the same number of parts. The triangles thus formod 
\\\\\ t('i-ni('(i the c'lcnicniary triangles of the polygon. 

Proh. XLVIIL — To inscribe amj regular polygon in a given circle ; or io 
divide the circumference into a given numler of equal j^rts^ hy means of 
the angle at the centre (fig. 141). 



GEOMETRICAL PROBLEMS. 



Suppose tlie circle is to contain a hexagon, or is to be divided at the 
circumference into six equal parts. Find the angle 
at the centre for a hexagon, or G0° ; draw any ra- 
dius B C, and set off by a protractor or otherwise 
the angle at the centre C B D, equal to G0° ; then 
the interval C D is one side of the figure, or seg- 
ment of the circumference ; and the remaining 
points of division may be found either by stepping 
alono; the circumference with the distance C D in 
the dividers, or by setting off the remaining five 
angles of 60° each round the centre. 




THE TJSE OF THE J SQUARE AKD TRIANGLE IN THE CONSTRUCIION OF SOME 
OF THE FOREGOING PROBLEMS. 



From the description of the T square it may be seen, that by sliding 
the stock along two contiguous edges of the board, the left hand and 
bottom edges, any number of parallel and perpendicular lines may be 
drawn. In so far, therefore, the T square supersedes the application of all 
the problems for drawing parallels and perpendiculars, coinciding in 
direction with the edges of the board ; for the square need only be set 
with its edge coincident with the points through which the line is to be 
drawn, and the pen or pencil drawn along the edge will describe the line 
required. When the perpendiculars or upright lines are of short length, 
the triangle and ruler are used. For this purpose, the triangle or set 
square of 60° is preferable to that of 45°, as it is longer and lighter. 

When the lines to be drawn do not coincide in direction with the edges 
of the board, the square may be adjusted with its bevel stock to the 
obliquity required, and the lines may be drawn as before. This is proba- 
bly the best plan when the oblique lines are numerous or extensive. In 
most cases, however, oblique lines are only occasional, and when their 
position is given, they may be drawn with a straight-edge. Wlien the 
oblique parallels and perpendiculars are short, as in oblique framing, short 
rods or bars, bolt-heads, and the like, the com- 
bined use of the straight-edge and triangle 'is 
expedient. Square figures may be described 

on a given centre, at one setting of the / 

straight-edge, as in the drawing of the head Fig. 142. 

of a square nut n (fig. 142). From the given centre, with a radius equal 




64: 



GEOMETEiCAL PROBLEMS. 



to lialf tlie side of tlie square, describe a circle, and with the aid of the tri- 
angle draw lines tangent to four sides. 

To draw an octagon, apply the set-sqnare of 45° to the corners, after 
completing a square figure, and di-aw tangents to the inscribed circle, as, 
for example, the line A k (fig. 138). 

To draw an equilateral triangle upon a given line A B (fig. 143), it is 
only necessary to apply the slant edge of the set-square of 60° to each end 






Fis. 143. 



Fig. 144. 



Fiff. 145, 



of the base, with the short side h c applied to the square-blade, and to 
draw the two sides A C, B C. If the given side A B be upright (fig. 144), 
apply the long side a h to the straight-edge, and draw as before. 

To draw a regular hexagon about a circle, with two of its sides parallel 
to the lower edge of the board : draw the centre line A B (fig. 145), and 
the upper and lower sides D E, F G, touching the circle, and apply the 
triangle of 60° touching the circle for the four remaining sides, as shown 
in the figure. 

When the hexagons are to be inscribed in the circle, first draw the 
centre line A B (fig. 133) as a diameter, and from the ends A, B, with the 

set-square, draw four sides cutting the circle 
|\ at D, E, F, G, and join D E, F G. 

i \ The triangles of 45° and 60° are useful in 

setting out the centre-lines of wheels with 3, 
4, 6, 8, (fee, arms, by drawing lines through 
\ the centre of the wheel. To set out 12 spokes 

.^i.L .^^s|£ }p in a wheel (fig. 146) : — Draw two diameters, 

A B, CD, parallel to the two edges of the 
board ; in the quadrant A C, draw radii E <?, 
E Z>, with the long and tlie short sides of the 
triangle against the square-blade. Tliose will 
divide tlie quadrant equally ; and the same 
construclii.n l.cing employed for the other quarters of the circle, 12 centre 




'lu'. IJO. 



GEOMETEICAL PEOBLEMS. 



C5 



lines, equally distant, will be described. Should tlie triangle be large 
enongli to embrace tlie wliole circle at once, tlie opposite quadrants A C 
and B D may be divided with the same setting of the triangle. 

A short method of dividing a line or surface into a number of equal 
parts is illustrated by fig. 147 ; and it is convenient where an ordinary 

rule does not evenly /i jr j^ 

measure the dimen- 
Suppose the 



sion. 

width A C is to be di- 
vided into seven equal 
parts, and that it mea- 
sures 7 1 inches ; an or- 
dinary inch rule, it is 




plain, does not afford the subdivisions when applied directly ; but if 14 
inches of length, or double the number of parts, be applied obliquely across 
the space between the parallels A B, C D, so as to measure it exactly, 
then point off two-inch intervals on the edge of the rule, and in this way 
7 equal subdivisions will be effected, through which parallels may be 
drawn. 



SIMPLE APPLICATIONS OF EEGULAE FIGUEES. 



Peob. XLIX. — To cover a surface with equilateral triangles^ hexagons^ 
or lozenges. 

Describe an equilateral triangle 
AB C, and produce the sides in- 
definitely. Set off from one angle 
A, equal intervals at «, &, a\ l>\ 
&c., as required ; and through 
these points draw parallels to each 
of the sides of the triangle. The 
area will be covered with triangles 
as required. 

For hexagons, or equilateral 
triangles and hexagons on the 
same surface, or lozenges, group the triangles. 

Peob. L. — To cover a surface loitJi octagons and squares. 

Draw two straight lines A B, A C, at right angles ; set off equal inter- 

5 




ee 



GEOMETRICAL PEOBLEMS. 



vals A d, d e, &c., on each line, equal to the breadth of the octagon to be 
described, and through these points draw parallels to the given lines, to 




^£ 



form sqnares. Within these sqnares constrnct octagons, by Probs. XLIII. 
or XLIY., and finish as in the figure. 



PROBLEMS ON PROPORTIONAL LINES AND EQUIVALENT FIGURES. 

Prob. LI. — To divide a given straight line into two jparts projoortional 
to two given lines. 

Let A B (fig. 160) be the line to be divided ; di-aw the straight line 
A D at any angle with A B, and set off A E, E D, equal to the other two 
given lines. Join D B, and draw C E parallel to it ; this line divides A B 
at C in the required ratio. 

Prob. LII. — To divide a straight line into any mimher of jparts of given 
proportions ; or simila/rly to a given straight line. 

Let A B (fig. 151) be the line to be divided. Draw B G at any angle 




J) 



Cy 



n T 

Fi- IM. 




wilh it, and scl. oil' by any convenient scale, B 0, D, <kc., to G, respec- 
tively, eqnal lo the given divisions. Join A G, and from the points of 



GEOMETRICAL PROBLEMS. 



G7 



division of B G draw parallels to A G, cutting it at II, I, &c. Tlie paral- 
lels so drawn will divide A B as required. 

Pkob. Lin. — To find a fourth proportional to three given lines. 

Draw two lines I K, I N" (fig. 152), at any angle, and set off I M, I N, 
equal to the two first of tlie given lines, and set off I L equal to the third. 
Join L IT, and draw ^N" K parallel to it. Then I K is a fourth proportional 
as required. 

The tvo first lines may be set ofiP successively on the same line, as from 
I to M, and from M to I^, and the third from I to L ; then L K will be 
tlie fourth line required. 

Prob. LIY. — To find a mean 2yroj[)ortional hetween two given lines (fig. 
153). 

Let a h and 5 <? be the given lines. Set off, on a straight line, A B, B C, 
equal to the given lines ; bisect A C at D, and with D A as a radius describe 
the semicircle A E C ; draw B E perpendicular to A C, meeting the circle 
at E. Then B E is the mean proportional required. 



b h C 



—~~.E 



Fig. 153. 




Pkob. LY. — To constmwt a triangle eqical in area to a given rectangle. 

Bisect the base B C (fig. 154) of the rectangle at D, and di-aw the per- 
pendicular D A equal to twice the height, D E, of the rectangle. Draw 
B A, A C ; the triangle A B C is equal in area to the rectangle B G. 

Peob. LYI. — To construct a square equal to a given rectangle (fig. 155). 

Let A B C D be the rectangle ; 
produce A B, and set off B E equal 
to the side B C of the rectangle ; bi- 
sect A E at K, and describe a semi- 
circle on A E ; draw the perpendicu- 
lar B H, cutting the circle at H, and 
on B H describe the square B G 
requu'ed. 

Peob. LYn. — To construct a triangle equivalent to any regular polygon. 

Eind the radius of the circle inscribed in the polygon. Set off on a 



K 



Fig. 155. 




68 



GEOMETRICAL PEOBLEMS. 



right line a distance equal to half the snm of the sides of the polygon. 
Tliis distance will be the base of the equivalent triangle, and the radius of 
the inscribed circle its perpendicular or altitude. 



PEOBLEMS ON THE ELLIPSE, THE PAUABOLA, THE HTPEEBOLA, THE CYCLOID, 

A:ND THE EPICYCLOID. 

The Ellipse. 

Peob. LYIII. — To describe an ellijpse, the length and hreadth, or the two 
axes being given. 

1st Method (fig. 156). — Bisect the transverse axis A B at C, and through 




C draw the pei-pendicular D E, making C D and C E each equal to half 
the conjugate diameter. On D as a centre, with C A as radius, describe 
arcs cutting at F, F^, for the foci. Divide A C into a number of parts at 
the points 1, 2, 3, &c. With radius A 1 on F and F^ as centres, describe 
arcs ; and with radius B 1, on the same centres, describe arcs inter- 
secting the others as shown. Repeat the operation for the other divisions 
of the transverse axis. The series of intersections thus found Avill be 
points in the curve, and they may be as numerously found as is desir- 
able ; after which a curve traced through them will form the complete 
ellipse. 

2^7 Mrfhnd (iig. 1 57).— llie two axes, A B, I) E, being given. On A B 
and I) K as dianieters iVoni the same centre C, describe circles F G, II I; 
take a convenient number of 2K)ints, a, />, Sec, m the scMni-circumferencc 
A 1M5, and (h-a\v radii cutting the inniuM- circle at a' 7/, c'vrc. ; from (7, h, 
t^'c, draw perpendieubirs to A B, and from a\ h\ Szv., draw parallels to 



GEOMETRICAL PEOBLEMS. 



G9 



A B, cutting tlie respective perpendiculars at n, o, &c. Tlie points of in- 
tersection so fonnci are points in tlie cnrve. 

Bd Ifethod (fig. 
158). — Along tlie 
straiglit edge of a slip 
of stiff paper mark off 
a distance a e equal 
to A^C, half tlie trans- 
verse axis ; and from 
the same point, a dis- 
tance a 1) equal to 
C D, half the conju- 
gate axis. Place the 
slip so as to bring the 
point 1) on the line 
A B of the transverse 
axis, and the point g 
on the line D E ; and 
set off on the drawing 
the position of the point 
a. Always keeping the 
point 5 on the transverse 
axis, and the point c on 
the conjugate axis, any q 
required number of points 
may be found. 

Uh Method (fig. 159). 

—By the above method 
large curves may be de- 
scribed continuously by 
means of a bar m h^ with 

steel points m, Z, Tc^ riveted 

into brass slides, adjusted to 

the length of the semi-axes, 

and fixed with set-screws. 

A rectangular cross E G, 

with guiding slots, is placed 

to coincide with the two 

axes of the ellipse A C and B H ; by sliding the points k, I, in the slots. 




YO 



GEOMETEICAL PROBLEMS. 



and carrying round the point m, tlie curve may be completely described. 
If desirable, of course, a pen or pencil may be fixed at m. 

Uh Method (fig. 160). — Given tlie two axes A B, C D ; on the centre 

C, with A E as radius, describe 
an arc cutting A B at F and G, 
the foci ; fiiL a couple of pins 
into the transverse axis at F 
and G, and loop on a thread 
or cord upon them, equal in 
B length, when looped on, to A 
B, so as, when stretched, as per 
dot-line FOG, just to reach 
the extremity, C, of the conju- 
gate axis. Place a pencil or 
draw-point inside the cord, as 
at H, and guiding the pencil 
in this way, keeping the cord equally on tension, pass round the two 
points F, G, and describe the curve as required. 

Tliis method is employed in setting off elliptical garden-plots, walks, 
&c. 

Peob. LIX. — To draw a tangent to an ellipse through a given point 
in the curve (fig. 161). 

From the given point T draw straight lines to the foci F, F' ; produce 





F T beyond the curve to c, and bisect the exterior angle c T F' by the line 
T d. Tliis line T c? is the tangent required. 

Piion. LX. — To draw a tangent to an ellipse from a given point lolth- 
oat the curre (iig. 162). 

Ki-oiii \hv given point T as centre, with a radius equal to its distance 
fi-oni \]\o, iK'ar(>st locus F, describe an arc; from the other focus F', with 
llie Iransverse axis as radius, cut the arc at K, L, and draw K F', L F', 



GEOMETEICAL PROBLEMS. 



Tl 



cutting the curve at M, ^; then the lines T M, T ^tsT, are tangents to the 
curve. 

T) 




Pkob. LXI. — To describe an ellipse approximately^ hy means of circular 



arcs. 



First, with arcs of two radii (fig. 163). Take tlie difference of the trans- 
verse and conjugate axes, and set it off from the centre O to <^ and (?, on 
O A and O C ; draw a c^ and set off half a do d; draw d i parallel to a <?, 




set off ^ equal to O cZ, join e i, and draw e m, d m, parallels to d i, i e. On 
centre m, with radius m C, describe an arc thi'ough C, and from centre i 
describe an arc through D ; on centre d, <?, also, describe arcs through A 
and B. The four arcs thus described form approximately an ellipse. This 
method does not aj)ply satisfactorily when the conjugate axis is less than 
two-thirds of the transverse axis. 

Second, with arcs of three radii (fig. 164). On the transverse axis A B, 
draw the rectangle B G, equal in height to C, half the conjugate axis. 
Draw G D pei-pendicular to A C ; set off O K equal to O C, and on A K 
as a diameter, describe the semicircle A 1^ K ; draw a radius parallel to 
O C, intersecting the semicircle at N and the line G E at P ; extend O G 



72 



GEOMETRICAL PROBLEMS. 



to L and to D ; set off O M equal to P N, and on D as a centre, witli a 
radins D M, describe an arc ; from A and B as centres, with a radius O L, 




I) 

Fig. 164. 



intersect this arc at a and h. The points H, a, D, h, W, are the centres of 
the arcs required ; produce the lines a H, D a, T) h, h H^, and the spaces 
enclosed determine the lengths of each arc. 

Tliis process works well for nearly all proportions of ellipses. It is 
employed in striking out vaults and stone bridges. 



The Parabola. 

Tlie parahola may be defined as an ellipse whose transverse axis is in- 
finite ; its characteristic is that every point in the curve is equally distant 
from the directrix E ]N" and the focus F (fig. 165). 

Prob. LXII. — To construct a jparohola lolien the focus and directrix are 
given. 

\st Method (fig. 1G5). — ^Tln-ough the focus F draw the axis A B perpen- 
dicular to the directrix E N, and bisect A F at e^ tlien e is the vertex of the 
curve. Tlirough a scries of points C, D, E, on the directrix, draw parallels 
to A B ; connect these points C, D, E, witli the focus F, and bisect by 
perpend icuhirs the linos F C, F D, F E. Tlie intersections of these per- 
pciidlciilars \s\\\\ \\w ])arjillcls will give points in the curve C'D'E^ through 
whicli trace tlie jiarabola. 

Id Method (iig. 1()C;).— Place a straight edge to the directrix E N, and 



GEOMETRICAL PROBLEMS. 



73 



apply to it a square LEG; fasten at G one end of a cord, equal in length 
to E G ; fix tlie other end to the focns F ; slide the square steadily along 




Fig. 166. 



the straight edge, holding the cord taut against the edge of the square by 
a pencil D, and it will describe the curve. 

Pkob. LXIII. — To construct cc parabola token the vertex A, the axis A B, 
and a point M of the curve are gwen (fig. 167). 

Construct the rectangle A B M C ; divide M C into any number ot 

equal parts, four for instance ; di- a - ^ - --■?-.- --/ -^ JX 

vide A C in like manner ; con- 
nect A 1, A 2, A 3 ; through V 
2' Z\ draw parallels to the axis. 
The intersections I, II, III, of 
these lines are points in the re- 
quired curve. 

Pkob. LXIY. — To draw a tangent to a given ]}oint II of the parabola 
(iig. 167). 

From the given point II let fall a perpendicular on tlie axis ^i h / ex- 
tend the axis to the left of A ; make A a equal to A 5 / draw a II, and 
it is the tangent required. 

The lines perpendicular to the tangent are called normals. To find 
the normal to any point I, having the tangent to any other point II. — Draw 
the normal II c / from I let fall a perpendicular I d on the axis A B ; lay 
QiS. d e equal to h c ; connect I e, and we have the normal required. Tlie 
tangent may be drawn at I by a perpendicular to the normal I e. 




74 



GEOMETKICAL PROBLEMS. 



The Hyperbola. 

An hyperlola is a curve from any point P in which, if two straight 
lines be drawn to two fixed points, F F' the foci, their difference shall 
always be the same. 

Peob. LXY. — To describe an hyperbola (fig. 168). 

From one of the foci F, with an assumed radius, describe an are, and 
from the other focus F^, with another radius exceeding the former by the 
given difference, describe two small arcs, cutting the first as at P and^. 
Let this operation be repeated with two new radii, taking care that the 
second shall exceed the first by the same difference as before, and two new 
points will be determined ; and this determination of points in the curve 
may thus be continued till its track is obvious. By making use of the 
same radii, but transposing, that is, describing with the greater about F, 
and the less about F^, we have another series of points equally belonging 
to the hyperbola, and answering the definition ; so that the hyperbola con- 
sists of two separate branches. 




Fig. 168. 



Fig. 169. 



The curve may be described mechanically (fig. 169). — By fixing a ruler 
to one focus F', so that it may be turned round on this point, connect the 
extremity of the ruler E to the other focus F by a cord shorter than the 
wliolc h'lio-lh V \X of the ruler by the given difference ; then a pencil P 
keci)ing tliis cord always stretched, and at tlie same time pressing against 
the edge of the ruler, will, as the ruler revolves around F', describe an 
liyperbohi, of which F F' arc the foci, and tlie diftorenccs of distances from 
these points to every point in tlie curve will be the same. 

]*i:()i;. LXVI. — To draw a tangent to any point P of an hyperbola ^fig. 



GEOMETPwICAL PK0BLEM3. 



iO 




On r' P lay off P^ equal to F P ; connect F^;>, and from P let fall a 
peiiDendicnlar on this line F^;>, and 
it will be the tangent required. 

The three curves, the ellipse, 
the parabola, and the hyperbola, 
are called conic sections, as they 
are formed by the intersections of a 
plane with the surface of a cone 
(plate Y). 

If the cone be cut through both 
its sides by a plane not parallel to 

the base, the section is an ellipse ; fis- ito. 

if the intersecting plane be parallel to the side of the cone, the section is a 
parabola ; if the plane have such a position, that when produced it meets 
the opposite cone, the section is a hyperbola. Tlie oj)posite cone is a 
reversed cone formed on the apex of the other by the continuation of its 
sides. 

The Cycloid. 

The cycloid is the curve described by a point in the circumference of a 
circle rolling on a straight line. 

Peob. LXYII. — To describe a cycloid (fig. 171). 

Draw the straight line A B as the base ; describe the generating circle 
tangent to the centi-e of this line, and through the centre C draw the line 
E E parallel to the base ; let fall a perpendicular from C upon the base ; 



.m 


5LX ;+' 


;?' 


3l2 \r • 


i 


1 \ T^ \ 




4 k 




•M 


i 

1 


AAAyyi 



E 



A ">' 4' 3' 2' 1' J) 

Fig. 171. 



B 



divide the semicircumference into any number of equal parts, for instance 
six ; lay off on A B and C E distances C 1', 1' 2' . . ., equal to the divisions 
of the circumference ; draw the chords D 1, D 2 . . ., from the points 1', 2', 3' . . . 
on the line C E, with radii equal to the generating cii'cle, describe arcs ; 



76 



GEO^klETEICAL PEOBLEMS. 



from tlie points 1^ 2^, 3', 4', 5' on the line B A, and with radii eqnal snc- 
cessively to the chords D 1, D 2, D 3, D 4, D 5, describe arcs cutting the 
preceding, and the intersections will be points of the curve required. 

2^ MetJiod (fig. 172).— Let 
9' be the base line, 4 9 the 
half of the generating circle ; 
divide the half circle into any 
number of equal parts, say 9, 




7\ ^S\ 



and draw the chord 1, 2, 
3, &c. ; lay off on the base 
V, V 2^, 2^ 3\ . . . , equal re- 
spectively to the length of one 
of the divisions of the half 
circle 1 ; draw through the 
points 1^, 2', 3' lines par- 
allel to the chords 1, 2, 
3 .... ; the intersections I, 
. of these lines are centres of the arcs a^ al^l c , of which 



Tt' 



TEL'' 



II, III . 

the cycloid is composed. 



The Epicycloid. 

The epicycloid is formed by a point in the circumference of a circle 
revolving either externally or internally on the circumference of another 
circle as a base. 

Prob. LXYIII. — To clescrihe an epicycloid. 

Let us in the first place take the exterior curve. Divide the circum- 
ference A B D (fig. 173) into a series of equal parts 1, 2, 3 ... ., beginning 
from the point A ; set off in the same manner, upon the circle A M, A 1^, 
the divisions 1', 2', 3' ... . equal to the divisions of the circumference A BD. 
Then, as the circle A B D rolls upon the circle A M A IST, the points 1, 2, 3 
will coincide successively with the points 1', 2', 3' ; and, drawing radii 
iVoiH the point O throngli the points 1', 2', 3', and also describing arcs of 
circles from the centre O, through the points 1, 2, 3, . . . ., they will inter- 
sect each other successively at the points c^ d^ e Take now the dis- 
tance 1 to r^ and set it off on the same arc from the point of intersection, ^, 
of tlie radius A C ; in like maimer, set off the distance 2 to d, from h to A*, 
and llie distance 3 to e to A", and so on. Tlien the points A\ A', A', 
will be so many ])oints in the (epicycloid ; and their frequency may be in- 



GEOMETKICAL PROBLEMS. 



77 



creased at pleasure by sliortening the divisions of the circular arcs. Tlius 
the form of the curve may be determined to any amount of accuracy, and 
completed by tracing a line through the points found. 

As the distances 1 to c, ... . wliich are near the commencement of the 
curve, must be very short, it may, in some instances, be more convenient 
to set oiF the whole distance ^ to 1 from c, and in the same way the distance 
h to 2 from d to A", and so on. In this manner the form of the curve is 
the more likely to be accurately defined. 




2d Method. — ^To find the points in the curve, find the positions of the 
centre of the rolling circle corresponding to the points of contact 1^, 2^, S\ 
&c., which may be readily done by producing the radii from the centre O, 



through the points 1\ 



.... to cut the circle B C. From these centres 



describe arcs of a circle with the radius of C A, cutting the corresponding 
arcs described from the centre O, and passing through the points A^, A-, 
A^, .... as before. 

When the movino; circle A B D is made to roll on the interior of the 
circumference A M, A X, as shown (fig. ITtt), the curve described by the 
point A is called an inteyior ejpicijcloid. It may be constructed in the 



78 



GEOMETEICAL PEOBLEMS. 



same way as in the preceding case, as may be easily imderstood, the same 
figures and letters of reference being used in both figures. 




1^-^-7 



Fig. 174. 



Tlie Involute. 

Tlie involute is a curve traced by the extremity of a flexible line un- 
winding from the circumference of a circle. 



h ':i 




Piioij. LXIX. — To doicribe an involute. 

Dividetlie circumference of the given circle (fig. 175) into any mmibcr 



GEOMETRICAT. PRORLE^IS. 



79 



of equal parts, as 0, 1, 2, 3, 4, . . . . ; at each of these points draw tangents to 
the given circle ; on the first of these lay off the distance 1 1^, equal to the 
arc 1; on the second lay off 2 2', equal to twice the arc 1 or the arc 
2 : establish in a similar way the points 3', 4', 5', .... as far as may be 
requisite, which are points in the curve required. 

It may be remarked, that in all the problems in which curves have 
been determined by the position of points, that the more numerous the 
23oints thus fixed, the more accurately can the curve be drawn. 

The involute curve may be described mechanically in several ways. 
Thus, let A (fig. 176) be the cen- ^ 
tre of a wheel for which the fonn 
of involute teeth is to be found. 
Let m n a he a thread lapped 
round its circumference, having 
a loop-hole at its extremity cCj' 
in this fix a pin, with which de- 
scribe the curve or involute a h 
.... A, by unwinding the thread 
gradually from the circumfer- 
ence, and this curve will be the proper form for the teeth of a wheel of the 
given diameter. 




The Spiral. 

The spiral is the involute of a circle produced beyond a single revolution. 

Peob. LXX. — To describe a spiral (figs. 5 and 6, plate XII). 

Divide the circumference of the j)rimary into any number of equal 
parts, say not less than eight. To these points of division e^f, ^, (fee, di-aw 
tangents, and from these points draw a succession of circular arcs ; thus, 
from 6 as a centre, with the radius e g^ equal to the arc a e reduced to a 
straight line, describe the arc a g h; from /", with the radius hf^ describe 
the arc g h ; from i the next arc, and so on. Continue the use of the centres 
successively and repeatedly to the extent of the revolutions required. 
Tims the point a in the fig. is used as a centre for three arcs, 'bl^om^ d n. 



80 GEOMETEICAL PEOJECTION. 



GEOMETEICAL PEOJECTIOI^. 

Aechitecttjeal and meclianical drawing is generally the delineation of 
bodies by geometrical or orthographic projection ; the representation, on a 
sheet of paper which has only two dimensions length and breadth, of solids 
which have three, length, breadth, and thickness. 

Since the surfaces of all bodies may be considered as composed of 
points, the first step is to represent the position in space of a point, by re- 
ferring it planes whose position is established. The projection of a point 
upon a plane is the foot of the perpendicular let fall from the point on the 
plane. If, therefore, on two planes not parallel to each other, whose posi- 
tions are known, we have the projections of a point, the position of this 
point is completely determined by erecting perpendiculars from each plane 
at the projected points : their intersection will be the point. 

If from every point of an indefinite straight line A B (fig. 177), placed 
j\ in any manner in space, perpendiculars be let 

fall on a plane L M E" O, whose position is given, 
then all the points in which these perpendiculars 
meet the plane will form another indefinite straight 
line a h : tins line is called the projection of the 
line A B on this plane. Since two points are 
sufficient to determine a straight line, it is only 
necessary to project two points of the line, and 
the Btraiglit line drawn through the two projected points will be the pro- 
jection of the given line. The projection of a straight line, itself perpen- 
diciihir to tlie plane, is the point in which this perpendicular meets the 

If tlic ]>r()jections ah and r// 7>' of a straight line on the two planes 
L M X () and L M V Q (tig. 178) are known, this line A B is determined ; 
for if, tlirongli oik^ of its projections « Z*, we suppose a plane drawn perpen- 



GEOMETRICAL PROJECTION. 



81 



dicularly to LMNO, and if through a' V another plane be drawn perpen- 
dicular to L M P Q, the intersection of the two planes will be the line A B. 
To delineate a solid, as the form of a machine for instance, it must be 
referred to three series of dimensions, each of them at right angles to the 
plane of the other. d zp^ f 




Fig. 178. 



Fig. 1T9. 



„.g. 



Thus, let ah G (fig. 179) be a parallelopiped in an upright position, of 
which the plane <^ 5 is horizontal, and the planes a c and c h vertical. Let 
d e^ df^ and d g^ be the boundary planes of a cubical space in which the 
body al c is j)laced ; the sides of the body being parallel to those planes, 
each to each, let the figure of the parallelopiped be projected on these 
planes ; for this purpose draw parallel lines from the angles of the body 
perpendicular to the planes, as indicated by the dotted lines ; then upon the 
plane d e we shall have a' V^ the projection of the surface a h : this is called 
the plan of the object. Upon the plane df^ve have a^ g\ the projection 
of the surface a c, tliQ front elevatio7i / and upon the j^lane dg, the projec- 
tion h' g' of the surface 5 g^ the side elevation. Here, then, we have three 
distinct views of the regular solid ah g delineated on plane surfaces, which 
convey an accurate and sufficient idea of its form. Indeed, any two of 
these representations are sufiicient as a description of the object. From 
the two figures a' g\ h' g\ for example, the third figure a' V may be com- 
pounded, by merely drawing the vertical lines g' A, h' i^ and a' Z;, g' l^ to 
meet the plane d <?, and by producing them horizontally till they meet and 



82 



GEOMETRICAL PROJECTION. 



form the figure a' V . Similarly, the figm-e V c' may be deduced from the 
other two by the aid of the lines A, % from a' h\ and the lines m, n^ from 



a G 




It is in this way that a third view of any piece of machinery is to be 
found from two given views ; and in many cases two elevations, or one 
elevation and a plan, may afford a sufficiently complete idea of the con- 
struction of a machine. In other cases, many parts may be concealed by 
others in which they are enclosed ; this suggests the occasional necessity 
of views of the interior, in w^hich the machine is supposed to be cut across 
by planes, vertically or horizontally, so as properly to reveal its structure. 
Such views are termed sections^ and, with reference to \hQ planes of section^ 
are denominated vertical and horizontal sections. To all such draw^ings 
is given the general title of geometrical drawings, as distinguished from 
perspective drawings. 

By the aid of drawing instruments, measurements are transferable from 

one position to another ; and there 
is no necessity for erecting three 
such planes as are supposed in fig. 
1Y9, upon which to execute draw- 
ings of a machine. In practice, 
the drawings are done upon one 
common surface, and we may readi- 
ly suppose the plane d g moved 
back into the position d g\ and d e 
also moved to d e' , both of these 
positions being in the plane of df. 
This being done, we have the three 
views depicted on one plane surface 
(fig. 180). In this figure, the same 
letters of reference are employed 
as in fig. 179 ; dl and d m are the 
ground and vertical lines. It is 
evident that the positions of the 
same i)oints in a' c' and a' V are in the same perpendicular from the ground 
line : that, in short, the position of a point in the plane may be found by 
ap])lying the edge of the square to the same point as represented in the 
elevation. The Hanic remark is applicable as between the two elevations. 
Hence tlic m(^tliod of drawing several views of one machine upon the same 
Buriacc of paper in strict agreement with each other. 



-k \i 




Fig. 180. 



GEOMETRICAL PROJECTION. 



83 



OF SHADE LINES. 



In outline drawings^ or drawings which consist simply of the lines em- 
ployed to indicate the form of the object represented, the roundness, the 
flatness, or the obliquity of individual surfaces, is not indicated by the 
lines, although it may generally he inferred from the relation of different 
views of the same part. The direct significance of an outline drawing may, 
however, be considerably increased, by strengthening thoge lines which in- 
dicate the contours of surfaces resting in the shadow ; and this distinction 
also improves the general appearance of the drawing. Tlie strong lines, to 
produce the best effect, ought to be laid upon the sharp edges at the sum- 
mits of salient angles ; but bounding lines for curve surfaces should be 
drawn finely, and should be but slightly, if at all, strengthened on the 



Fig. 181. 



Fig. 183. 



Fig. 185. 





f\ 



e-\ 




Fig. 182. 



Fig. 184. 



Fig. 186. 



shade side. This distinction assists in contrasting flat and curve surfaces. 
To understand and apply the shade lines, however, we must know the 
direction in which the light is supposed to fall upon the object, and thence 
the locality of the shadows. 

It is necessary for the explicitness of the drawing, that firstly, the light 
be supposed to fall upon the object in parallel lines, that all the parts may 



84: GEOMETRICAL PEOJECTIO]^. 

be shade-lined according to one uniform rule ; secondly, that the light 
should be supposed to fall upon the object obliquely, as in this way both 
the horizontal and vertical lines may be relieved by shading. To distribute 
the shadows equally, the light is supj)osed to fall in directions forming an 
angle of 45° with both the horizontal and the vertical planes of projection. 
In general, the light should fall, as it were, from towards the upper left- 
hand corner of the sheet of paper, supposing it square, making also an 
angle of 45° with the surface. 

To illustrate what has been stated, let ab c d and a' h' ef (figs. 181 and 
182) represent the elevation and plan of a solid rectangular body, JNT O 
being the ground line. Let the direction of the light in both views be 
represented in projection by the arrows A B ; these lines form the angle 
45° with the line 1^ O, and by drawing the parallels at 5, d^ a\ e\ so as to 
embrace the extreme contour, we may readily perceive the way in which 
the light falls upon the body : it falls upon three faces, namely, the two 
vertical faces a' f, f e^ and the top a' V ef. Consequently, the intersec- 
tions or lines at which these planes meet ought to be lightly drawn, namely, 
a 5, ad; a' f 2jcA f e. Again, the lateral planes represented by ^ c, c d^ 
y g\ and a! V^ are obviously in the shade, as no light falls upon them 
directly ; and these lines are strengthened to express the distinction. 

In figs. 183 and 184, the portion of the exterior from h by c Xo d\% in 
the shade, while the rest is light ; and the inverse is the case with the inner 
edges. A peculiarity, however, occurs at cZ, for here the edges, inner and 
outer, are parallel to the direction of the light. It is plain that the surfaces 
which come up to these edges will be in a medium shade, and that the 
lines at d should be of medium thickness. 

Figs. 185 and 186 represent a hollow cylinder in projection. In the 
plan, two lines, «, <?, drawn parallel to the direction of the light, and touch- 
ing the exterior of the cylinder, define the semicircular outline a V c, which 
is thrown in the shade, and ought to be strengthened. The outlines a and 
c are, like the edges at d (fig. 183), parallel to the light, and the contour 
on each side gradually recedes and advances to the light. Tlie thickness 
of the line should, tlicrefore, be rather gradually reduced at the points a, c. 
In tlie elevation, the base-line rZ/ should be shaded, and h d k often half- 
shaded, as it lies in a curve surfiice ; more generally full-shaded. 

If, again, tlic cylinder be hollow, presenting in plan the interior contour 
circle e/i^ tlicn the semicircle e(j h expresses the shady side of the interior, 
the light striking directly upon the o'j)|:>posite semicircle. 

These exam])lcs ilhistrate every case of shade-lining tliat occurs in out- 
line drawings. The effect is enhanced by proportioning the thickness of 



GEOMETRICAL PROJECTION. 



85 



the lines to the depth of the surfaces to which they belong, below the 

^ Fig. 1S7. 



ori<rinal surfaces from which the shadows 



arise. 

In the later French system of shading, 
the light is supposed, in plan, to strike to- 
wards the right hand upper corner, falling, 
as it were, in front of the objects ; but in 
elevation, towards the right hand and foot of 
the sheet (figs. 188, 187). 

It will be observed in the illustrations of 
this work, that in the tinted drawings, the 
shadow is thrown according to the French 
system ; that is, the light is supposed to fall 
on the drawing over the left shoulder at an 
angle of 45°. But in outline drawings, on 
account of its greater simplicity, the more 
usual system of throwing the shade line one way, both in plan and eleva- 
tion, is adopted. 




Fiff. 188. 



PROJECTIONS OF SIMPLE BODIES. 

Projections of a regular hexagonal pyramid (PL I, II).— It is evident 
that two distinct geometrical views are necessary to convey a complete 
idea of the form of the object : an elevation to represent the sides of the 
body, and to express its height ; and a plan of the upper surface, to ex- 
press the form horizontally. 

It is to be observed that this body has an imaginary axis or centre-line, 
about which the same parts are equally distant ; this is an essential charac- 
teristic of all s}niimetrical figures. 

Draw a horizontal straight line L T through the centre of the sheet ; 
this line will represent the ground line. Then draw a perpendicular Z 2! 
to the ground line. For the sake of preserving the symmetry of the draw- 
ing, the centres of the lower range of figures are all in the same straight 
line M N, drawn parallel to the ground line. 

Figs. 1, 2. — In delineating the pyramid, it is necessary, in the first 
place, to construct the plan. The point S', where the line Z Z' intersects 
the line M N, is to be taken as the centre of the figure, and from this 
point, with a radius equal to the side of the hexagon which forms the base 
of the pyramid, describe a circle, cutting M N in A' and D\ From these 



86 GEOMETRICAL PROJECTION. 

points with the same radius, draw four arcs of circles, cutting the primary 
circle in four points. These six points being joined by straight lines, will 
form the figure A' B^ C^ D' E^ F^, w^hich is the base of the pyramid ; and 
the lines A^ T>\ B^ E^, and C F^, will represent the projections of its edges 
fore-shortened as they w^ould appear in the plan. If this operation has 
been correctly performed, the opposite sides of the hexagon should be 
parallel to each other and to one of the diagonals ; this should be tested 
by the application of the square or other instrument proper for the pur- 
pose. 

By the help of the plan obtained as above described, the vertical pro- 
jection of the pyramid may be easily constructed. Since its base rests 
upon the horizontal plane, it must be projected vertically upon the ground 
line ; therefore, from each of the angles at A', B^, 0\ and D', raise per- 
pendiculars to that line. The points of intersection. A, B, C, and D, are 
the true positions of all the angles of the base ; and it only remains to 
determine the height of the pyramid, which is to be set off from the point 
G to S, and to draw S A, S B, S C, and S D, which are the only edges of 
the pyramid visible in the elevation. Of these it is to be remarked that 
S A and S D alone, being parallel to the vertical plane, are seen in their 
true length ; and moreover, that from the assumed position of the solid 
under examination, the points F^ and E^ being situated in the lines B B' 
and C C^, the lines S B and S C are each the projections of two edges of 
the pyramid. 

Figs. 3 and 4.— T<9 construct the projections of the same jpyrarriidy hav- 
ing its base set in an inclined jposition^ l)ut with its edges S A and S D still 
parallel to the vertical ^lane. 

It is evident, that with the exception of the inclination, the vertical 
projection of this solid is precisely the same as in the preceding example, 
and it is only necessary to copy fig. 1. For this purpose, after having 
fixed the position of the point D upon the ground line, draw through this 
point a straight line D A, making with L T an angle equal to the desired 
inclination of the base of the pyramid. Then set off the distance D A, fig. 
1, from D to A, fig. 3 ; erect a j)erpendicular on the centre, and set ofiF 
G S equal to the height of the pyramid. Transfer also from fig. 1 the dis- 
tance B G and C G to the corresponding points in fig. 3, and complete the 
figure by (h-awing the straight lines A S, B S, C S, and D S. 

In constructing the ])lan of the pyramid in this position, it is to be re- 
marked, tliat since the edges S A and S D are still parallel to the vertical 
plane, and tlic. ])oint D remains unaltered, the projection of the point A 
Avill si 111 be in llie line M T^. Its position at A' (fig. 4) is determined by 



PLATE Y 



90 




GEOMETRICAL PROJECTION. 91 

tion as fore-sliortened, and not in its true dimensions, we sliall now proceed 
to the consideration of tlie second question proposed. Let the cutting 
plane X X be conceived to turn upon the point J, so as to coincide with 
the vertical line h k^ and (to avoid confusion of lines) let hhhQ transferred 
to a' 1)\ which will represent, as before, the extreme limits of the curve 
required. Now, taking any point, such as f/, it is obvious that in this new 
position of the cutting plane, it will be represented by cZ', and if the cutting 
plane were turned upon a' V as an axis till it is parallel to the vertical 
plane, the point which had been projected at d? w^ould then have described 
round a' V an arc of a circle, whose radius is the distance d' d^ (fig. 2). 
This distance, therefore, being set oif at d^ and f on each side of a' h\ 
gives tw^o points in the curve sought. By a similar mode of operation any 
number of points may be obtained, through which, if a curve be drawn, it 
wdll be an ellipse of the true form and dimensions of the section. 

Figs. 3 and 4. — To find the horizontal projection and actual outline of 
the section of a cone^ made hy a plane Y Y parallel to one side or generatrix^ 
and perpendicular to the vertical plane. 

Determine by the second method laid down in the preceding problem 
any number of points, as F^, G^, J^, K^, &c., in the curve representing the 
horizontal projection of the section specified. The horizontal plane pass- 
ing through M gives only one point M^ (which is the vertex of the curve 
sought), because the circle which denotes the section that it makes with 
the cone is a tangent to the given plane. 

In order to determine the actual outline of this curve, suppose the 
plane Y Y to turn as uj)on a pivot at M, until it has assumed the position 
M B, and transfer M B parallel to itself to M' B'. Tlie point F will thus 
have first described the arc F E till it reaches the point E, which is then 
projected to E'; suppose the given plane, now represented by M^ B^, to 
turn upon that line as an axis, until it assumes a position parallel to the 
vertical plane, the point E^, which is distant from the axis M^ B' by the 
distance F' S^ (fig. 4), wdll now be projected to F' (fig. 3). Tlie same dis- 
tance F^ S' set ofiT on the other side of the axis M^ B^ gives another point 
G^ in the curve required, which is XhQ parabola. 

Figs. 7, 8, 9. — To draw the vertical projection of the sections of two op- 
posite cones made hy a plane parallel to their axis. 

Let C E D and C B A be the two cones, and X X the position of the 
cutting plane (fig. 7). Project in plan either of the cones, as in fig. 8 ; 
from its centre, with a radius equal to L H, describe . a circle, and draw 
the tangent ha; ha will be the horizontal projection of the cutting plane. 
Draw the line H^ M^ (fig. 9) parallel to the cutting plane ; H'', M' corre- 



92 GEOMETRICAL PROJECTION. 

sponding in position to the intersections H, M (fig. 7), of the plane with the 
cones. From H^ and M^ lay off distances eqnal to L K, K I, and the length of 
the cone, and through these points draw perpendiculars, as/' e' ^ d' c' ^ V a', 
(fee, which must be made equal to the cliords/^, dc^h a (fig. 8), made by 
the cutting plane a h, with circles whose radii are G K, I F, and the radius 
of tlie base of the cone. Through the points a\ c\ e\ H',/*', d', h\ draw 
the curve, and we have the projection required. A similar construction 
will give the sectional projection of the opposite cone at M^ The curve 
thus found is the hyperbola. 



PENETRATIONS OR INTERSECTIONS OF SOLIDS. ' 

On examining the minor details of most machines, we find numerous 
examples of cylindrical and other forms, fitted to, and even appearing to 
pass through each other in a great variety of ways. The examples grouped 
in plates YI. to XI. are selected with the view of exhibitijig those cases 
which are of most frequent occurrence, and of elucidating general principles. 



PENETRATIONS OF CYLINDERS. 

Plate YI. — Figs. 1 and 2 represent the projections of two cylinders of 
unequal diameters meeting each other at right angles ; one of which is 
denoted by the rectangle A B E D in the vertical, and by the circle 
A' ir B' in the liorizontal projections ; while the other, which is supposed 
to be horizontal, is indicated in the former by the circle L P 1 1^, and in 
the latter by the figure L' I' K' M^ From the position of these two solids 
it is evident that the curves formed by their junction will be projected in 
the circles A' II' B' and L P I E" ; and further, that such would also be 
the case even although their axes did not intersect each other. 

But if the position of these bodies be changed into that represented at 
figs. 3 find 4, tlie lines of their intersection will assume in the vertical pro- 
jection a totally different aspect, and may be accurately determined by 
the following construction. 

Throiigli any ])()iiit taken u])()n the j^lan (fig. 4) draw a liorizontal line 
a' h\\v]\\v\\ is to \)v considciH'd as indicating a i)lanc cutting both cylinders 
parallel to tlicir axes ; this plane would cut the vertical cylinder in lines 
drawn perpendicnlarly through the points c/ and d'. To find the vertical 
projection of its intersection with the other cylinder, conceive its base PL', 



PLATE y I 



92 




« S i i i 


be 




■"I; ; 




/li 




^ i V 


\ : r^ 




— S-. '^ 

V 1 ; 


•^ 


■^ \^^ 1 ^^^ ~K, 


■^ 


C: 


^^^-^ 



PLATE Y IL 




I , 

t 







GEOMETRICAL PROJECTION. 93 

after being transferred to I" L', to be turned over parallel to the horizontal 
plane ; this is expressed by simply drawing a circle of the diameter I^ L^ ; 
and producing the line a' V to ci^ ; then set off the distance a^ e' on each 
side of the axis I K, and draw straight lines through tliese points parallel 
to it. These lines a J, g A, denote the intersection of the j^lane a' V with 
the horizontal cylinder, and therefore the points c, cZ, m, o^ where they cut 
the perpendiculars c c\ d d\ are points in the curve required. By laying 
down other planes similar to a' l>\ and operating as before, any number of 
points may be obtained. Tlie vertices i and Ic of the curves are obviously 
projected directly ; and their extreme points are determined by the inter- 
sections of the outlines of both cylinders. "When the cylinders are of 
unequal diameters, as in the present case, the curves of penetration are 
hyperbolas. 

Figs. 5 and 6. — When the diameters of the cylinders are equals and 
when they cut each other at right angles, the curves of penetration are pro- 
jected vertically in straight lines perpendicular to each other, as in fig. 5, 
where the projections of some of the points are indicated in elevation and 
plan by the same letters of reference. 

Figs. 7 and 8. — To delineate the intersections of two cylinders of equal 
diameters at right angles^ when one of the cylinders is inclined to the Tier- 
tical plane. 

Supposing the two preceding figures to have been drawn, the projec- 
tion c of any point such as c' may be ascertained by observing that it must 
be situated in the perpendicular c' <?, and that since the distance of this 
point (projected at c in fig. 5) from the horizontal plane remains unaltered, 
it must also be in the horizontal line c c. Upon these principles all the 
points indicated by literal references in fig. 7 are determined ; the curves 
of penetration resulting therefrom intersecting each other at two points 
projected upon the axial line L K, of which that marked q alone is seen. 
Tlie ends of the horizontal cylinder are represented by ellipses, the con- 
struction of which will also be obvious on referring to the figures ; and 
they do not require further consideration here. 



PENETRATIONS OF CYLINDERS, CONES, AND SPHERES. 

PI. YIII.,figs. 1 and 2. — To find the curves resulting from the inter- 
section, of two cylinders of unequal diameters^ meeting at any angle. 

For the sake of simplicity, suppose the axes of both cylinders to be 
parallel to the vertical plane, and let A B E D and JST O Q F be their pro- 



94: GEOMETEICAL PEOJECTION". 

jections upon that plane. In constructing, in the first place, their horizon- 
tal projection, observe that the upper end A B of the larger cylinder is 
represented by an ellipse A' K' B' M^, which may easily be drawn by the 
help of the major axis K^ M' equal to the diameter of the cylinder, and of 
the minor A' B', the projection of the diameter. The visible portion of 
the base of the cylinder being similarly represented by the semi-ellipse 
L' D' H', its entire outline will be completed by drawing tangents L' M' 
and H^ K^ The upper extremity P J!!^ of the smaller cylinder will also be 
projected in the ellipse ^^ i' W . 

Now, suppose a plane, as a' g' (fig. 2), to pass through both cylinders 
parallel to their axes ; it will cut the surface of the larger cylinder in two 
straight lines passing through the points/' and g' on the upper end of the 
cylinder ; these lines will be represented in the elevation, by projecting 
the points/"' and g' to/ g ; and drawing (^/and c g parallel to the axis. 
The j)lane a' g' will in like manner cut the smaller cylinder in two straight 
lines, which will be represented in the vertical projection by <^ A and e % 
and the intersections of these lines with af and c g will give four points 
Z, ^', m, and n^ in the curves of penetration. Of these points one only, that 
marked Z, is visible in the plan, where it is denoted by V . 

Fig. 1.^ — To find the curves of jpenetration in the elevation without the 
aid of the plan. 

Let the bases D E and Q O of both cylinders be conceived to be turned 
over into the vertical plane after being transferred to any convenient dis- 
tance, as D^ E^ and Q^ O'^, from the principal figure ; they will then be 
represented by the circles D^ H^ E^ and Q^ G' O^. E'ow draw a'^c^ paral- 
lel to D E, and at any suitable distance from the centre I ; this line will 
represent the intersection of the base of the cylinder with a plane parallel 
to the axes of both, as before. The intersection of this plane with the 
base of the smaller cylinder will be found by setting oflP from K a distance 
R^?, equal to I ^, and drawing through the point p a straight line parallel 
to Q O. It is obvious that the intersection of the supposed plane with the 
convex surfaces of the cylinders will be represented by the lines af c g^ 
and d A, e ^, drawn iiarallel to tlie axes of the respective cylinders through 
the points where tlic chords a"^ c^ and d^ e"^ cut the circles of their bases ; 
and tliat, consequently, the intersections of those lines indicate points in 
the curves sought. These points may be multiplied indefinitely by con- 
ceiving other ])l;ines to pass through the cylinders, and operating as 
before. 

Figs. ^3 and 4. — To find the ctirvcs of penetration of a cone and 
sphere. 



PLATE V I I ] 




PLATE IX 



94 





OEOilETKICAL PROJECTION. 95 

Let D S be the axis of the cone, A' L' B' tlie circle of its base, and the 
triangle A B S its projection on the vertical plane ; and let C, C, be the 
projections of the centre, and the circles E' K' F' and E G F those of the 
circnmferences of the sphere. 

Tliis problem, like most others similar to it, can be solved only by the 
aid of imaginary intersecting planes. Let a h (fig. 3) represent the pro- 
jection of a horizontal plane ; it will cut the sphere in a circle whose diam- 
eter is a h, and which is to be drawn from the centre C in the plan. 
Its intersection with the cone is also a circle described from the centre S' 
with the diameter c d; the points e^ and/"', where these two circles cut 
each other, are the horizontal projections of tAVO points in the lower curve, 
which is evidently entirely hidden by the sphere. The points referred to 
are projected vertically upon the line ah at e and f. The upper curve, 
which is seen in both projections, is obtained by a similar process ; but it 
is to be observed that the horizontal cutting planes must be taken in such 
positions as to pass through both solids in circles which shall intersect each 
other. For our guidance in this respect it wdll be necessary, first, to de- 
termine the vertices m and 7i of the curves of penetration. 

For this purpose, conceive a vertical plane passing through the axis of 
the cone and the centre of the sphere ; its horizontal projection will be the 
straight line C L' joining the centres of the two bodies. Let us also make 
the supposition that this plane is turned upon the line C C as on an axis, 
until it becomes parallel to the vertical plane ; the points S^ and L' will 
now have assumed the positions S^ and JJ, and consequently the axis of 
the cone will be projected vertically in the line D' S^, and its side in S^ L^, 
cutting the sphere at the points p and ?\ Conceive the solids to have 
resumed their original relative positions, it is clear that the vertices or 
adjacent limiting points x)f the curves of penetration must be in the hoii- 
zontal lines ^ o and r q, drawn through the points determined as above ; 
their exact positions on these lines may be ascertained by projecting ver- 
tically the points m' and ?i^, where the arcs described by the points j9 and 
r, in restoring the cone to its first position, intersect the line S L. 

It is of importance further, to ascertain the points at which the curves 
of penetration meet the outlines A S and S B of the cone. Tlie plane 
which passes through these lines being projected horizontally in A' B', will 
cut the sphere in a circle whose diameter is i^f; this circle, described in the 
elevation from the centre C, will cut the sides A S and S B in four points 
at which the curves of penetration are tangents to the outlines of the cone. 

Figs. 5 and 6. — To find tlie lines of penetration of a cylinder and a 
cylindrical ring or tones. 



96 GEOMETRICAL PKOJECTION. 

Let the circles A^ E' B', F' G' K\ represent the horizontal, and the 
figure A C B D the vertical projection of the torus, and let the circle 
H^f L^, and the rectangle H I M L be the analogous projections of the 
cylinder, which passes perpendicularly through it. Conceive, as before, 
a plane a h (fig. 5), to pass horizontally through both solids ; it will ob- 
viously cut the cylinder in a circle which will be projected in the base 
H^/^ L' itself, and the ring in two other circles, of which one only, part 
of which is represented by the arc /^ h^ V ^ will intersect the cylinder at 
the points y and h^^ which being projected vertically to fig. 5, will give 
two points/* and h"^ in the upper curve of penetration. 

Another horizontal plane, taken at the same distance Ijelow the centre 
line A B as that marked ah \'s> above it, will evidently cut the ring in 
circles coinciding with those already obtained ; consequently the points/' 
and h^ indicate points in the lower as well as in the upper curves of pene- 
tration, and are projected vertically at d and e. Thus, by laying down 
two planes at equal distances on each side of A B, by one operation four 
points in the curves required are determined. 

To determine the vertices m and n^ following the method explained in 
the preceding problem, draw a plane O n\ passing through the axis of the 
cylinder and the centre of the ring, and conceive this plane to be moved 
round the point O as on a hinge, until it has assumed the position O B', 
parallel to the vertical plane ; the point n\ representing the extreme 
outline of the "cylinder in plan, will now be at r' ^ and being projected ver- 
tically, that outline will cut the ring in two points^ and r^ which would 
be the limits of the curves of penetration in the supposed relative position 
of the two solids ; and by drawing the two horizontal lines r n and j9 m^ 
and projecting the point n' vertically, the intersections of these lines, the 
two points m and n^ are the vertices of the curves in the actual position of 
the penetrating bodies. 

The points at which the curves are tangents to the outlines II I and L M 
of tlie cylinder, may readily be found by describing arcs of circles from 
the centre O tln-ough the points H' and L^, which represent tliese lines in 
the plan, and then proceeding, as above, to project the points thus obtained 
upon the elevation. Lastly, to determine the points, as ^;, z^ &c., where 
the curves are tangents to the horizontal outlines of the ring, draw a circle 
P' s' j' witli a radius equal to that of the centre line of the ring, namely, 
P D ; the points of intersection 3' and / are the horizontal projections of 
the ])oints sought. 

liequired to rcjpresent the sections which woidd he made in the ring noiv 
"before 7/,v, hy livo planes^ one- of which^ "N' T', is j)araU('l to the vertical 



GEOMETEICAL PROJECTION. 97 

2)lane^ wJiile the otlier^ T E', is jperpendicular to hoth jplanes of prelec- 
tion. 

Tlie section made by the last-named plane mnst obviously have its ver- 
tical projection in the line C D, which indicates the position of the plane ; 
but the former will be represented in its actual form and dimensions in 
the elevation. To determine its outhnes, let two horizontal planes g ^ and 
i A", equichstant from the centre line A B, be sujDposed to cut the ring ; 
their lines of intersection with it will have their horizontal projections in 
the two circles g' o' and h' q' which cut the given plane W T in o' and q\ 
These points being projected vertically to o^ q^ k^ &c., give four points in 
the curve requii'ed. The Hne W T cutting the circle A' E' B' at W, the 
projection jST of this point is the extreme limit of the curve. 

The circle P' s' j\ the centre line of the rim of the torus, is cut by the 
planes X' T' at the point s\ which being projected vertically upon the 
lines D P and C Z, determines s and I, the points of contact of the cmwe 
with the horizontal outlines of the ring. Finally, the points t and ii are 
obtained by di-awing from the centre O a cu-cle T' v' tangent to the given 
plane, and projecting the point of intersection v' to the points v and x, 
which are then to be replaced upon C D by drawing the horizontals v t 
and X u. 



PE^sETEATIOXS OF CTLIXDEES, PEISMS, SPHEEES, AND CONES. 

Plate X., figs. 1 and 2. — Required to delineate the lines of penetra- 
tion of a sphere and a regular hexagonal prism whose axis passes through 
the centre of the sphere. 

The centres of the cii'cles forming the two projections of the sphere are, 
according to the terms of the problem, upon the axis C Q>' of the upright 
prism, which is projected horizontally in the regular hexagon D' E^ E^ G' 
H' r. Hence it follows, that as all the lateral faces of the prism are equi- 
distant from the centre of the sphere, their lines of intersection with it will 
necessarily be circles of equal diameters. JN'ow, the perpendicular face 
represented by the line E' E' in the plan, will meet the surface of the 
sphere in two circular arcs E F and L M (fig. 1), described from the centre 
C, with a radius equal to c' V or a' c' . And the intersections of the two 
oblique faces D'E' and E' G' will obviously be each projected in two arcs 
of an ellipse whose major axis dg\^ equal to the diameter of the circle acl)^ 
and the minor axis is the vertical projection of that diameter, as represented 
at e' f (fig. 2). But as it is necessary to draw small portions only of these 
curves, the following method may be employed. 



98 GEOMETRICAL PROJECTION. 

Draw D G ; throiigli the points E, F, divide the portions E F and F G 
respectively into the same luimber of equal parts, and drawing perpen- 
diculars through the points of division, set off from F G- the distances from 
the corresponding points in E F to the circular arc EOF, as points in the 
elliptical arc required. The remaining elliptical arcs should be traced by 
the same method. 

Figs. 3 and 4. — Heqidred to draw the lines of penetration of a cylinder 
and a sphere^ the centre of the sphere heing without the axis of the cylinder. 

Let the circle Y>' YJ \I be the projection of the base of the given cylin- 
der, the elevation of which is shown at fig. 3, and let A B be the diameter 
of the given sphere. If a plane, as c' d\ be drawn parallel to the vertical 
plane, it will evidently cut the cylinder in two straight lines G G^, H H', 
parallel to the axis, and projected vertically from the points G' and H^ 
This plane will also cut the sphere in a circle whose diameter is equal to 
d d\ and which is to be described from the centre C with a radius of half 
that line ; its intersection with the lines G G^ and H H^ will give so many 
points in the curves sought, viz., G, H, I, K. 

The planes a' V and e' f ^ which are tangents to the cylinder, furnish 
only two points respectively in the curves ; of these points E and F alone 
are visible, the other two, L and M, being concealed by the solid ; there- 
fore, the planes drawn for the construction of the curves must be all taken 
between a' V and e' f . The plane which passes through the axis of the 
cylinder cuts the sphere in a circle whose projection upon the vertical 
plane will meet at the points D, IS", and (/, 7^, the outlines of the cylinder, 
to which the curves of j)enetration are tangents. 

Figs. 5 and 6. — To find the lines of penetration of a truncated cone 
ajid a prism. 

The straight line C D is the axis of a truncated cone, which is repre- 
sented in the plan by two circles described from the centre G ; and the 
horizontal lines M ]^ and M^ W are the projections of the axis of a prism 
of which the base is square, and the faces respectively parallel and per- 
pendicular to the planes of projection. 

In laying down the plan of this solid, it is supposed to be inverted, in 
ordw^that the smaller end of tlic cone, and tlic lines of intersection of the 
lower surface F G of the prism may be exhibited. According to this 
arrangement, the letters A' and B' (fig. G) ought, strictly speakhig, to be 
marked at tlie points I' and IF, and conversol}'' ; but as it is quite obvious 
tliat th(; ])art al)(»ve W W is exactly synnnetrical witli that below it, the 
(listi'ihulioii ol' tlic letters of reference adopted in our iigures can lead to no 
coni'iisioii. 



\ 



r J. A T E X . 




r L A T E X 1 




/ 






PLATE XI 



!M 








GEOMETRICAL rROJECTION. 



The intersection of the plane F G with the cone is projected horizon- 
tally in a circle described from the centre C, with the diameter F' G'. 
The arcs V F^ A' and 11^ G' B' are the only parts of this circle which re- 
quire to be drawn. 

Figs. 7 and 8. — To describe the curves formed hj the intersection of a 
cylinder with the frustum of a cone^ the axes of the txoo solids cutting each 
other at right angles. 

The axes of the solids and their projections are laid down in the figures 
precisely as in the preceding example. The intersections of the outlines 
of the cone in the elevation with those of the cylinder, furnish, obviously, 
fom- points in the cmwes of j)enetration ; these points are all projected 
horizontally upon the line A! B'. I^ow, suppose a plane, as co h (fig. 7), 
to pass horizontally through both solids ; its intersection with the cone 
will be a circle of the diameter c d^ while the cylinder will be cut in two 
parallel straight lines, represented in the elevation by cc ^, and whose hori- 
zontal projection may be determined in the following manner : — Conceive 
a vertical plane /*^, cutting the cylinder at right angles to its axis, and let 
the circle g ^/"thereby formed be described from the intersection of the 
axes of the two solids ; the line^ h will now represent, in this position of 
the section, the distance of one of the lines sought from the axis of the 
cylinder. E'ow set off this distance on both sides of the point A', and 
through the points Ic and a' thus obtained, draw straight lines parallel to 
A' B' ; the intersections of these lines with the circle drawn from the 
centre C of the diameter c d wall give four points m\ p\ n, and (?, which 
being projected vertically upon cc ^, determine two points m and ^9 in the 
curves required. 

In order to obtain the vertices or adjacent limiting points of the curves, 
draw from the vertex of the cone a straight line t <?, touching the circle g ef 
and let a horizontal j^lane be supposed to pass through the point of con- 
tact e. Proceed accordino- to the method 2:iven above to determine the 
intersections of this plane with each of the solids in question, the four 
points i\ r' ^ ^, and 5, which being projected vertically upon the line e r, 
determine the vertices i and r required. 



OF THE HELIX. 



Plate Xn. — The Helix is the curve described upon the surface of a 
cylinder by a point revolving round it, and at the same time moving 
parallel to its axis by a certain invariable distance dming each revolution. 
Tliis distance is called the pitch of the screw. 



100 GEOMETRICAL PEOJECTION. 

Figs. 1 and 2. — Required to construct the helical curm described hy the 
point A \ipon a cylinder ^projected horizontally in the circle k! Q' F', the 
pitch being represented hy the line A' A^ 

Divide tlie pitch A' A^ into any number of equal parts, say eight ; and 
through each point of division, 1, 2, 3, &c., draw straight lines parallel to 
the ground line. Then divide the circumference A^ C^ F^ into the same 
number of parts ; the points of division B^, C^, E^, F^, &c., will be the 
horizontal projections of the different positions of the given point during 
its motion round the cylinder. Thus, when the point is at B' in the plan, 
its vertical projection will be the point of intersection B of the perpendicu- 
lar di'awn through B^ and the horizontal drawn through the first point of 
division. Also, when the point arrives at C in the plan, its vertical pro- 
jection is the point C, where the perpendicular drawn from C^ cuts the 
horizontal passing through the second point of division, and so on for all 
the remaining points. The curve A B C F A^ drawn through all the 
points thus obtained, is the helix required. 

Figs. 1 and 2. — To draw the mrtical elevation of the solid contained 
between tioo helical surfaces and two concentric cylinders. 

A helical surface is generated by the revolution of a straight line 
round the axis of a cylinder ; its outer end moving in a helix, and the line 
itself forming with the axis a constant and invariable angle. 

Let A^ C^ F^ and 1\! W O' represent the concentric bases of the cylin- 
ders, whose common axis S T is vertical ; the curve of the exterior helix 
A C F A^ is the first to be drawn according to the method above shown. 
Then having set off from A to A^ the thickness of the required solid, draw 
through A^ another helix equal and similar to the former. ISTow construct, 
as above, another helix, K C O, of the same pitch as the last, but on the 
interior cylinder ; as also another, K^ C^ O'^, equal and parallel to the 
former. The lines A' K', B^ L', C W^ &c., represent the horizontal pro- 
jections of the various positions of the generating straight line, which, in 
the present example, has been supposed to be horizontal ; and these lines 
are projected vertically at A K, B L, e%c. 

It will be observed, tliat in the position A K the generating line is 
projected in its actual length, and that at the position C W its vertical 
projection is the point C. The same remark applies to the generatrix of 
tlie scc(>nd helix. The parts of both curves which are visible in the ele- 
vation may be easily determined by inspection. 

Figs, o and 4. — To determine the vertical projection of the solid fonned 
by a sphere 'inoving in a helical curve. 

Let A' Q' E' be the base of a cylinder, upon which the centre point C 



i^ L A T K X J 1 




GEOMETRICAL PKOJECTION. 101 

of a sphere whose radius is a' C describes a liclix, which is projected on 
the vertical pLane in the curve A C E J. After determining as above the 

various points A, B, C, D , in this curve, draw from each of these 

points as centres, circles with the radius a' C ; the circumferences of tliese 
circles will denote the various positions of the sphere during its motion 
round the cylinder ; and if lines be drawn touching these circles, the 
curves thereby formed will constitute the figure required. One of these 
curves will disappear at O, which is its point of contact with the circle 
described from the point E, the intersection of the helix with the perpen- 
dicular E E' ; it will again reappear at the point I when it becomes a tan- 
gent to the circle described from the point J in the prolongation of the 
line A A^ The exterior and interior circles (fig. 4) represent the horizon- 
tal projection of the solid in question. 

The conical helix differs from the cylindrical one in that it is described 
on the sm-face of a cone instead of on that of a cylinder ; but the construc- 
tion differs but slightly from the one described. By following out the same 
principles, helices may be represented as lying upon spheres or any other 
surfaces of revolution. 

In the arts are to be found numerous practical applications of the heli- 
cal curve, as wood and machine screws, geers, and staircases, the construc- 
tion of which will be still farther explained under their appropriate heads. 



102 THE DEVELOPMENT OF SURFACES. 



THE DEYELOPMEKT OF SUEFACES. 

The development of the surface of a solid is tlie drawing or unrolling 
on a plane the form of its covering ; and if that form be cut out of paper, 
it would exactly fit and cover the surface of the solid. Frequently in 
practice, the form of the surface of a solid is found by applying paper or 
thin sheet brass directly to the solid, and cutting it to fit. Tin and copper- 
smiths, boiler-makers, &c., are continually required to form from sheet 
metal forms analogous to solids ; to execute which they should be able to 
construct geometrically the development of the surface of which they are 
to make the form. 

The development of the surface of a plain cylinder is evidently but a 
plane sheet, of which the circumference is one dimension whilst its length 
is the other. 



THE DEVELOPMENT OF THE SUEFACE OF INTERSECTED CTLINDEKS. 

Plate XIII., figs. 1 and 2. — To draw the surface of a cylinder formed 
l)y the intersection of another equal cylinder^ as the Icnee of a stove ^i^e. 

Let A B C D be the elevation of the pipe or cyhnder. Above A B 
describe the semicircle A^ 4' B' of the same diameter as the pipe ; divide this 
semicircle into any number of equal parts, eight for instance ; through these 
points 1', 2', 3', &c., draw lines parallel to side A C of pipe, and cutting 
the line C D of the intersection of the two cylinders. Lay off A^^ B'' equal to 
the semicircle A' 4' B', and divided into the same number of equal parts ; 
tlirough these points of division erect perpendiculars to A''' B", and on these 
perpendiculars lay off the distances A" C'\ 1" V, 2'^ 2'', W' W\ and so on, 
corresponding to A C, 1 1, 2 2, 3 3, etc., in preceding figure. Tlirough the 

points C, V\ %'\ D'^, tlien draw connecting lines, and we have the 

developed surface required. It is to be remarked, that this gives but one 
half of the surface of the pipe, the other being exactly similar to it. 



V L A T E XIII 





" 




/ 








\' 


i{ 

1 

1 
1 


1 






\ 




THE DEVELOPMENT OF SURFACES. 103 

Figs. 3 and 4. — To develop the surface of a cylinder intersected hj an- 
other cylinder^ as in the formation of a "[ j^W^- 

Tlie construction is similar to the preceding, and as the same letters 
and figures are preserved relatively, the demonstration will be easily 
understood from the foregoing. 

Tlie development of the surface of a right cone (figs. 5 and 6). From C 
(fig. 6) as a centre, 'svitli a radius C A' equal to the inclined side A C of 
the cone (fig. 5), describe an arc of a circle A' B' A'' ; on this arc lay off 
the distance A' B' A''' equal to the circumference of the base of the cone ; 
connect A' C and C A!\ and A' B' A'' C is the developed surface re- 
quired. 

To develop the surface of afriistrum of a cone^ D A B E (fig. 5). 

On ^g, 6 develop the cut-ofiT cone C D E as in preceding construction, 
and we have A^ B' A^^ D'^ E^ D as the developed surface of the right 
frustrum. 

To develop the surface of a frustrum of a cone, when the cutting plane 
a h (fig. 5) is inclined to the hase. 

On A B the base describe the semicircle A 3' B ; divide the semicircle 
into any number of equal parts, six for instance ; from each point of divi- 
sion 1^, 2', 3^5 4^, 5', let fall perpendiculars to the base ; at 1, 2, 3, 4, 5, con- 
nect each of these last j)oints with the apex C. Divide now the arc A' B' 
(fig. 6) into six equal parts, or the arc A^ B' A''' into twelve ; each of these 
parts by the construction is equal to the arc A 1\ V 2' (fig. 5) ; connect 
these points of division with, the point C^ ; on C^ A^ (fig. 6) take C^ a' equal 
to C (^ of fig. 5, a being the point at which the plane cuts the inclined side 
of the cone ; in the same w^ay on C B', lay off C^ h' equal to C h. 

It is evident that all the lines connecting the apex C with the base, 
included within the two inclined sides, are represented as less than their 
actual length in fig. 5, and must be projected on the inclined sides to de- 
termine their absolute dimensions ; project, therefore, the points V\ 2'', 3'', 
4:'\ 5'', at which the cutting plane intersects the lines C 1, C 2, C 3, C 4, 
C 5, by drawing parallels to the base through these points to the inclined 
side C B^ On fig. 6 lay off C^ r\ C 2"% &c., equal to C V% C 2,^^^, &c. 

(fig. 5) ; connect the points a', V\ 9,"% V, a% and we have the 

developed surface a' A! B^ A!' a" V required. 

To develop the surface of a sphere or hall (figs. 189, 190). 

It is evident that the surface cannot be accurately represented on a 
plane surface. It is done approximately by a number of gores. Let CAB 
(fig. 189) be the eighth of a hemisphere ; on C D describe the quarter 
circle D A {? ; divide the arc into any number of equal parts, six for in- 



104 



THE DEVELOPMENT OF SURFACES. 



stance ; from the points of division 1, 2, 3, . . . let fall perpendiculars on 
C D, and from the intersections with this line describe arcs V 1% 2' 2''', 3^ 3", 

cutting the line B at 1% 2% S% ; on the straight line C D' (fig. 

190), lay off C^ D' equal to the arc D A c, with as many equal divi- 




Fig. 190. 



Fig. 1S9. 



sions ; then from either side of this line lay off V" 1"", 2'" 2"" . . . . D' B' 

equal to the arcs 1' 1", 2' 2", D B (fig. 189). Connect the points C\ 

V% 2"\ .... and C A' B' is the developed surface. 

It is to be remarked, that in the preceding demonstrations, the forms are 
described to cover the surface only ; in construc- 
tion, allowance is to be made for lap by the addi- 
tion of margins on each side as necessary. It 
is found difficult in the formation of hemispherical 
ends of boilers, to bring all the gores together at 
the apex ; it is usual, therefore, to make them, as 
shown (fig. 191), by cutting short the gores, and 
Fig. 191. surmounting the centre with cap piece. 




MECHANICS. 



105 



mecha:n'ics. 

The profession of an arcliitectural or meclianical dranghtsman should 
embrace not merely the mere copying of examples which may be furnished 
him, but also the designing of new edifices and machines, in which he may 
draw from the results of his own experience ; from good models, by col- 
lating suitable parts from divers designs ; or by the rules of mechanics, 
proportioning the parts according to the magnitude and direction of the 
strains to which they are to be subject, and the materials of which they are 
to be composed; introducing as much of ornament as the subject may 
require. 



1st Class. 



THE MECHANICAL POWERS. 

The simple machines or mechanical 
powers which enter as elements into the 
composition of all machinery are the 
lever^ the pulley^ and the inclined plane, 
to which some add the toggle-joint and 
the hydrostatic press. The lever em- 
braces the wheel and axle, and the in- 
clined plane the wedge and screw. 

It is usual to regard the lever as of 
three kinds, distinguished by the relative 
position of power P, weight "W, and ful- 
crum F. In the first the fulcrum is be- 
tween the power and the weight ; in the 
2d, the weight is between the power and 
the fulcrum ; and in the 3d, the power is 
between the weight and the fulcrum. 
This division is rather nominal than real; 
all applications of the lever may be re- 
solved under one general rule : that the 



^V 



"ST 




? 


2d. Class. 












w 


F 










▲ 




s 



106 MECHANICS. 

intermediate weight, pressure, or tension, is eqnal to the sum of the outside 
ones, and the comparative size of these last two depends on their position 
in reference to the first. Let the intermediate pressure be exerted through 

the means of a spring balance c. Fig. 

192, it will mark the sum of the 

weights a and 5. 

Calling X the distance from a to 

c, and y from c to h ; knowing any 
Fig- 192. thrcc of the four, a, h, x and y, the 

fourth may be found ; for a multiplied by x is equal to 5 multiplied by y 
{ax=hy). Thus, if 6^ be 6 lbs., x 5 ft., and y 10 ft., then 6 multiplied by 
5=: 30, is equal to 10 multiplied by h ; therefore T^,or 3 lbs., is equal to h, 
and the load at c is 6+3=9 lbs. 

As c divides the lever proportionally to the weights, and as it is the 
sum of the weights, knowing c, x, and y, a and h may be calculated. Thus, 
if the load at c be 100 lbs., and x 7 ft., and y 3 ft., then one of the weights 
will be j\ of c, or 70 lbs., and the other y^^) o^ 30 lbs, ; and, as the heavier 
weight is at the shortest end of the lever, h will be TO lbs., and a 30 lbs. 
« is to 5 as 3 to 7, but a? is to y as 7 to 3 ; therefore, the weights a and h 
are to each other as the opposite ends of the lever, as y io x j or the point 
c divides the lever in the inverse proportion of the weights. 

If either a or h be known, and the lengths of the lever x and y^ the 
load at G may be calculated directly, c is equal to either weight, a or J, 
multiplied by the whole length of the lever, divided by y or a?, the part of 

a X (x A- if\ 
the lever opposite the known weight, c = . It will be readily 

understood that a, h, or c may be considered at option, either fulcrum, 
weight, or power, according to the requirements of the mechanism. 

The Wheel and Axle. — If a weight, P, be suspended from the periphery 
of a wheel, fig. 193, whilst another weight, W, is suspended on the oppo- 
site side of a barrel or axle attached to the wheel, the principle of action is 
the same as that of the lever. F multiplied by its length of lever or radius 
ca of the wheel is equal to W multiplied by its length of lever or radius 
of the axle ch ; the axis c is the fulcrum. If a movement downward be 
communicated to F, as shown by the dotted line, a rotary motion is given 
to the wheel and axle ; the cord of F is unwound whilst that of W is 
wound up, but P is still suspended from a and W from h / the leverage, or 
distance from the fulcrum, of each is the san^e as at first. The wheel and 
axle is a lever of continuous and uniform action. Since the wheel has a 
larger circumference than the axle, by their revolution more cord will be 



MECHAlilCS. 



107 



unwound from the former than is wound up on the latter, P will descend 
faster thanW is raised, in the proportion of the circumference of the 
wheel to that of the axle, or of their radii ca to ._ 

,.IU,^ .'''' "'-, 

cb. "When P has reached the position (J») W 




'.0/ 



1'F] 



a 



i^' 



Fio. 193. 



Y 



will have reached l^\. If caho 4o times c h, then / 
P will have moved 4 times the distance that W \ 
has. The movement is directly as the length of ' 
the levers, or the radii of the points of suspension. 
It will be perceived, therefore, to move a large 
weight by the means of a smaller one, that the 
smaller must move through the most space, and 
that the spaces desciibed are as the opposite ends 
of the lever, or inversely as the weights. 
/^ It is the fundamental principle of the action of 
all mechanical powers, that whatever is " gained 
in power," as it is said, is lost in space travelled ; 
that, if a weight is to be raised a certain number 
of feet, the force exerted to do this must always 
be equal to the product of the weight by the 

height to which it is to be raised ; thus, if 200 lbs. are to be raised 50 ft., 
\ the force exerted to do this must be equal to a weight, which, if multiplied 
by its fall, will be equal to the product 200 x 50, or 10,000 ; and it is 
immaterial whether the force be a weight of 10,000 lbs. falling one ft., or 
1 lb. 10,000 ft. 

It is now common to refer all forces exerted to a unit of lbs. ft. ; that 
is, 1 lb. falling 1 ft. ; and the effect to the same imit of lbs. ft., 1 lb. 
raised 1 ft. Thus, in the example above, the force exerted or power is 
10,000 lbs. ft. falling ; the effect 10,000 lbs. ft. raised. In practice, the 
lbs. ft. of force exerted must always be more than the lbs. ft. of effect pro- 
duced ; that is, there must be some excess of the former to produce move- 
ment, and to overcome resistance and friction of parts. 

The measure of any force, as represented by falling weight, is termed 
the absolute power of that force; the resulting force, or useful effect for the 
purposes for which it is applied, is called the effective power. 

The Pulley. — The single fixed pulley, ^g. 194, consists of a single 
grooved wheel movable on a pin or axis ; called fixed, because the strap, 
through which the pin passes, is attached to some fixed object. A rope 
passes over the wheel in the groove ; on one side the force is exerted, and 
on the other the weight is attached and raised. It may be considered a 



108 



MECHAIsICS. 




© 



A 

2 

VI 



xvyvvw^xxxx^m^x^^^^^^ ^^^^^ ^^^ ^^^^ ^^ ^^^^^ diameters, or as a lever in 
which the two sides are equal, the pin being the ful- 
crum. P, the force exerted, must therefore be equal to 
the weight, "W, raised ; and, if movement takes place, 
W will rise as much as P descends. 

The fixed pulley is used for its convenience in the 

Q application of the force ; it may be easier to pull down 

than up, for instance; but the lbs. of force must be 
Fig. 194. equal to the lbs. of effect. The tension on the rope is 

equal to either the force or weight. 

Fig. 195 is a combination of a fixed pulley A, and movable pulley P. 

The simplest way to arrive at the principle of this combination is to con- 

v^^^^^\\\^^^\^^^^^\^\^^N^\^^^^^^^ sider its action. Let P be pulled down, say 2 feet ; 

the length of rope drawn to this side of the pulley 

must be furnished from the opposite side. On that 

side there is a loop, in which the movable pulley 

with the weight W attached is suspended. Each 

side of this loop, 2 and 3, must go to make up the 

2 ft. for the side or end 1. Cords 2 and 3 will 

therefore furnish each 1 ft. As these cords are 

Fig. 195. shortened 1 ft. the weight W is raised 1 ft., and as 

the movement of W is but 1 ft. for the 2 ft. of P, "W must be twice that of 

P, because the 2 lbs. ft. of P must equal lbs. ft. of "W". 

In the combination of pulleys. Fig. 196. Let P be pulled, say 3 ft., 
then this length of rope drawn from the opposite 
side of the pulley is distributed over the 3 cords 2, 
3, 4, and the weight "W is raised 1 ft; conse- 
quently, the weight W is 3 times that of P. Tlie 
cord 1 supports P, the cords 2, 3, 4, the weight W, 
or 3 times P ; consequently, the tension on every 
cord is alike. The same ro]3C passing freely round 
. ^ I ^ pulleys must have the same tension thronghout ; so 

t£j ^ that, to determine the relation of W to P, count the 

number of cords which sustain the weight. Thus 
in Fig. 197 the weight is sustained by 4 cords ; con- 
hfA sequcntly it is 4 times the tension of the cord, or 4 
FiQ. 197. times the force P. hi order not to confuse the 
cords, the pulleys are represented as in the figures ; but, in construction, 
the pulleys, or sheaves, are usually of the same diameter, and those in con- 
nection; as A and P), and C and D, run on the same pin. 



D 



fJl^y.'j'flUd 



TT{ 



J 



FlO. 190. 



MECHAiaCS. 



109 



The Inclined Plane. — To su2:)port a weight by means of a single fixed 
pulley, tlie force must be equal to the weight. Suppose the weight, 
instead of hanging freely, to rest upon an inclined plane h d, Fig. 198 ; if 
motion ensue, to raise the weight W the height al^ the rope transferred 
from the weight side of the pulley will be equal to 1) d, and P will have 
consequently fallen this amount; thus, if hd\>Q Q ft., and all ft., whilst 
W is raised 1 ft. P has descended 6 ft., and as lbs. ft. of power must equal 
lbs. ft. of effect, P will be | of TV ; and, by reference to the figure, P is to 
W as ah \& to h d^ or as the height of the incline is to its length. If the 



c 





M^mmmMM 



Fig. 198. Fig. 199. 

end of the plane d be raised, till it becomes horizontal, the whole weight 
would rest on the plane, and no force would be necessary at P to keep it 
in position ; if the plane be revolved on 5, till it becomes perpendicular, 
then the weight is not supported by the plane at all, but it is wholly 
dependent on the force P, and is equal to it. Between the limits, there- 
fore, of a level and a perpendicular plane, to support a given weight W, 
the force P varies from nothing to an equality with the weight. 

The construction, Fig. 199, illustrates the principle of the icedge^ which 
is but a movable inclined plane ; if the wedge be drawn forward by the 
weight P, and the weight W be kept from 
sliding laterally, the fall of P, a distance 
equal to ad^ will raise the weight W a 
heiofht ch. P will therefore be to W, as <?5 
is to ad. For example, if the length of the 
wedge ad be 10 ft., and the back ch 2 feet, 
then P will be to W, as 2 to 10, or X of it. 

Let the inclined plane ahd^ Fig. 198, be 
bent round, and attached to the drum A, 
Fig. 200, to which motion of revolution on 
its axis is given, by the unwinding of the turns of a cord from around its 
periphery, through the action of a weight P suspended from a cord passing 
over a pulley. If the weight W be retained in its vertical position, by the 
revolution of the drum it will be forced up the incline, and when the cord 
has unwound one-half turn from the drum, and consequently the weight P 




^! 



1 

A 



Fio. 200. 



no 



MECHAlSnCS. 




descended a distance ce equal to one-half the circumference of the drum, 

the weight "W has been raised to the height al)\yj 
•7^ the half revolution of the plane ; P must therefore 
be to W as al is to one-half the circumference. 
, Extend the inclined plane so as to encircle the 
(pjdrum, Fig. 201. The figure illustrates the 
mechanism of the screw, which may be considered 
as formed by wrapping a fillet-band or thread 
around a cylinder at a uniform inclination to the axis. In practice, the 
screw or nut, as the case may be, is moved by means of a force applied at 
the extremity of a lever, a complete revolution raises the weight the dis- 
tance from the top of one thread to the top of the one above, or the jpitch. 
If the force be always exerted at right angles to the lever. Fig. 202, the 
lever may be considered the radius of a wheel, at the 
circumference of w^hich the force is applied. Thus, if 
the lever be 3 ft. long, the diameter of the circle 
would be 6 ft., and the circumference 6 x 3.1416 or 
18 yVo ft., if the pitch be 1 mch, or y\ of a foot, then 
the force would be to the weight as jV is to 18.85. ; 
and if the force be 1 lb., the weight would be 
226.20 lbs. 

Parallel Forces. — If two horses be harnessed to a load, the effect is 
to draw a load equal to the sum of their forces exerted in a line opposed 
to the resistance of the load at the point where the whiffle-tree is attached 
to the load. The forces, both of exertion and resistance, act in parallel 
lines, and the resultant of the two forces, which is their sum, and counter- 
balances the third, must be applied at a point intermediate between, 
and distant from each of them, inversely as the forces exei'ted. The 
composition of two parallel forces acting in the same direction is to be 
solved as an example of a lever, with an intermediate fulcrum. The 
forces are represented by the weiglits, and the point of application of the 
resultant by the fulcrum, which is acted on, as if there were but a single 
force, equal to the sum of the two, opposed to it. 

The resultant, of any number of parallel forces, acting in one direction, 
is equal to their sum, acting in the same direction at some intermediate 
point; that is, the effect of all these forces is just the same, as if there were 
but one force, equal to their sum, acting at this point, and is balanced by 
an equal force acting in the opposite direction. This central point may be 
determined by finding the resultant, i. e., the sum, and the point of appli- 




FiG. 202. 



MECHANICS. 



Ill 




cation for any two of tlie forces, and then of other two, the resultants thus 
determined being again added together like simple forces. 

As parallel forces can be added together ; reciprocally, they can also be 
divided. The single force, acting intermediately, may be resolved into 
forces acting at the ends of a lever, whose siun, whatever their number, 
will be equal to the central force. 

Inclined Forces. — When two men of equal strength pull directly opposite 
to each other, the resultant is nothing. Let a third take hold of the centre 
of the rope, and pull at right angles to 
the rope; he will make an angle in the 
rope and the two others are now pulling in 
directions inclined to each other. Tlie less 
the force exerted at the centre, the less the 
flexure in the rope ; but when it becomes 

equal to the other two, the two, to balance it, must pull directly against it, 
bringing the ends of the rope together, and acting as parallel forces. Be- 
tween the smallest force and the largest, that can be exerted at the centre 
and maintain a balance or equilibrium, the ends of the rope assume all 
varieties of angles, which angles bear definite relations to the forces. 

Represent these forces by weights, as in Fig. 203 ; let P and W be the 
extreme forces, acting over pulleys, and tending to draw the rope straight, 
which the weight C prevents : to 
find what must be the weight of C 
to balance the others. Above the 
rope draw the lines cjp and cii\ 



parallel to the rope, from c lay off 

on cp ]3art3 of an inch, equal to i 

number of pounds or units of r^^ 

weight in P ; that is, if P be 5 

lbs., and TT 6 lbs., lay off cj) |, y\, 

or any fractions of an inch that 

may be the most convenient, say f ; and cio \Q)i an inch ; draw^<^ parallel 

to GiG^ and %oa parallel to C'j9, connect ac ; if the length ac be measured 

in Stlis of an inch, or whatever fractional parts may be adopted for the 

other weights, it will represent the weight of C in lbs. (in this case about 

5|- parts, or lbs.), and the direction in which C acts ; the work of the 

weights P and TV is to support C, and this would be done by a force equal 

to C, 5J- lbs., acting in a line ca directly above it. Therefore, the force 

opposed to the direction of C, and equal to it, will be the resultant of the 

two forces, P and TV actino; at an ano-le to each other. 




Fig. 20S 



112 



MECHAOTCS. 




s 


i 


J 


I 




/ 




\^ 



Fig. 204. 



As two forces may- 
be compounded, so cou- 
yersely one force may be 
resolved into two; tbus, 
let the weight P, Fig. 
204r, be supported by two 
^JL inclined rafters CA and 
C B. Eacn resists a part 
of the force exerted by 
the weight P. To find 
the force exerted against 
the abutments A and B, 
in the direction of CA 
and CB, draw c^A' par- 
allel to C A, cB' to C B, 
and cJ, a continuation of 
the line C P, the direction 
in which the weight P acts ; lay off c <^ from a scale of equal jDarts, a 
length which will represent the number of lbs., or whatever unit of weight 
there may be in the weight P; draw da parallel to cB^, and dh parallel 
to cA^; ca^ measured on the scale of equal parts adopted, will represent 
the lbs. or units of weight exerted against A in the direction of C A, and 
c h the lbs. or units of weight exerted against B in the direction of C B. 

This method of finding the resultant of two forces, or the components 
of one force, is called the pai'allelogram of forces. If two sides of a paral- 
lelogram represent two forces in magnitude and direction, the resultant of 
these two forces will be represented in magnitude and direction by the 
diagonal of the parallelogram, and conversely. 

The sum oi ac and cl \^ greater than ccl ; that is, the weight P exerts 
a greater force in the direction of the lines C A and CB, against A and 
B, than its own weight ; but the down pressure upon A and B is only equal 
to the weight of P and of the rafters ^vhich support it, which last, in the 
present consideration, is neglected. Resolve cl), the force acting on B in 
the direction of cB', into gh or ce the downward pressure, and eg or eh 
the liorizontal thrust on the abutment B, and ca into cf and /*«. To 
decompose a force, form a triangle w^ith the direction of the other forces, 
upon the line representing the magnitude and direction of the given force. 
cc rei)rescnt3 the weight on B, cf or de the weight on A ; cd^or ce •{■ de^ 
the whole weight P ; therefore, the weight upon the two abutments A and 
B is equal to the whole weight of P. 



MECHANICS. 



113 



Connect A' B', and extend the line cd to E. c ^ is to <? ^ as A^ E is to 
E B/ and as the weight is distributed on the abutments A and B in propor- 
tion to ed and c e, it will be also as B' E is to A^ E ; or the whole construc- 
tion may be considered a lever, in which the weight is suspended at E and 
distributed between the supports inversely as its distance from them, af 
and e h represent the horizontal thrust on the abutments A and B, but as 
af and b e are equal, the force tending to separate them, or to tear asunder 
A'B' if a tie-rod, would be represented by either of them. 

In the application of forces to a lever we have considered them parallel 
and perpendicular to the lever ; if they are inclined, they may be resolved 
into forces acting pei'pendicular to and in the direction of the lever, or 
they may simply be referred to the fulcrum, as the axis of a wheel and axle. 





Fig. 205. 

of which the perpendiculars let fall from the fulcrum upon the direction of 
the forces are the radii. Thus, if the levers, Eig. 205, be acted on by 
forces of which the direction is shown by the arrows, the leverage, as it is 
called, or effective length of arm at which they act, are the perpendiculars 
E a and F 5, on the directions of the forces. 

Centre of Gravity. — The action of weight, or its tendency 
downward to the earth, is called gravity. Let a mass P, Fig. 
206, be suspended by a cord, each particle of the mass is acted 
upon by gravity, like an innumerable number of threads pulling 
it downward, and all these parallel forces may be resolved into 
a single force opposed to the direction of the string, and equal 
the sum of all the forces or the weight. 

Suspend the mass C, Fig. 207, by a cord from P ; the line 




Fig. 206. 






Fig. 207. 



Fig. 208. 




IM 



MECHAOTCS. 




Fig. 210. 



P M will represent the direction of the resultant through the mass. Sus- 
pend the same mass again from Q, the resultant will now take the direc- 
tion Q E, and the two resultants have one point C, their intersection in 
common ; this point is called the centre of gravity ; all the weight may be 
supposed concentrated at this point. 

A body placed on a horizontal plane will fall over, unless the vertical 
line passing through its centre of gravity fall within its base ; thus, in Fig. 
208, the body will stand firmly, whilst in Fig. 209 it will fall over. Per- 
sons carrying loads, Fig. 210, adjust their position in- 
sensibly so that the vertical line from the common cen- 
tre of gravity, ^, of their own bodies G, and of their 
load H, should fall within the area bounded by their feet. 
The body is thrown to the side opposite the load. 

When bodies are of symmetrical forms and homo- 
geneous, the determination of the centre of gravity is 
the finding the centre of the figure. The centre of 
gravity of a triangle is in a line drawn from its summit to the middle of 
the opposite side, and at -J of its length from the base. 

The centre of gravity of any quadrilateral may be found by dividing it 
into two triangles, and finding the centre of gravity of these triangles, con- 
necting them by a straight line, and dividing this line into parts inversely 
proportional to the surfaces of the two triangles, the point of division 
being the centre of gravity of the figure. In the same 
way any polygon may be subdivided into triangles 
of which the centres of gravity may be found, and re- 
solved by connections with each other. 

The centre of gravity of the triangular pyramid, 
Fig. 211, is in the straight line AE, connecting the apex 
A with the centre of gravity of the base triangle BCD, 
and distant J of the length of the line A E from E. 
The centre of gravity of solids, which may be divided into symmetrical 
figures and pyramids, as for all practical purposes most may be, can be 
found by determining the centre of gravity of each of the solids of which 
it is compounded, and then compounding them, observing that each centre 
of gravity represents the solid contents of its own mass or masses of which 
it may be composed. The centre of gravity of bodies enclosed by more or 
less regular contours, as a ship for instance, is determined by dividing it 
into parallel and equidistant sections, finding the centre of gravity of each, 
and compounding tlicni into a single one. 




Fig. 211. 



MECHANICS. 



115 



FRICTION, AND THE LIMITING ANGLE OF RESISTANCE. 

Suppose a mass A (fig. 212) bo pressed upon another, B, by means of 
a force acting in a direction perpendicular to the common surface of the 
two bodies, and let a second force Q act also upon it in a direction paral- 
lel to this surface. Then, since the forces P and Q act in directions per- 
pendicular to each other, they manifestly cannot counteract one another, 
and it would be expected that the body should move in the direction of 
the second force. This, however, is not always the case ; except the force 
Q exceed a certain limit, no motion ensues. Some new force F, therefore, 
has been produced in the system, counteracting the force Q ; that force is 
called friction. It acts always in a direction parallel to the surfaces in 
contact, and is always for surfaces of the same nature the same fraction^ 
or part of the force P ly which these are pressed together^ w^hatever be the 
amount of that force, or whatever the extent of surfaces in contact. This 
fraction is called the coefficient of friction. Whilst it is thus the same for 
the same surfaces, whatever be the extent of the surfaces, or the force with 
which they are pressed together, it is different for different surfaces. 

Construct the parallelogram of forces P V" Q M {:^g. 213). V" M rep- 
resent the resultant of the two forces P and Q. The actual friction is 
always a certain given fraction of P acting parallel to the impressed sur- 
face. Take M Q' equal to this given fraction of P M, complete the paral- 
lelogram, and draw the diagonal P' M. Since, then, M Q^ represents the 
friction of the body upon the plane, or the force called into action by P M, 
which opposes the motion of the body ; since, moreover, Q M represents 



V\F 




Fig. 212. 



M Q' Q 

Fis. 213. 




the force tending to produce motion, it follows that the body will or will 
not move, according as Q M is greater or less ,than M Q', or as the angle 
P M P^' is greater or less than P' M P. The angle P' M P is called the 
limiting angle of resistance. It depends upon the coefficient of friction, 
and is therefore the same fov the same surfaces, whatever be the actual 
amoimt of the impressed force P. 

Hence it appears, that force impressed upon the surface of a solid body 
at rest, by the intervention of another solid body, will be destroyed, pro- 



116 



MECHANICS. 



Tided the angle wMcli the direction of that force makes with the perpen- 
dicular to the surface does not exceed a certain angle called the limiting 
angle of the resistance at that surface, and this is true, however great the 
force may be. Also, that if the direction of the impressed force lie with- 
out this angle, it cannot be sustained by the resistance of the surfaces in 
contact, and that this is true, however small the force may be. 

Suppose a heavy mass (fig. 214), whose centre of gravity is G, to be 
placed on an inclined plane A B. The whole pressure of the mass may 
be supposed to act in the direction of the vertical line G M, and this pres- 
sure will be just destroyed by the resistance of the surface of the plane 
when the angle G^ P Q, which G P makes with the perpendicular P Q, is 
equal to the limiting angle of resistance. A mass of any substance will, 
therefore, just be sustained on an inclined plane, without slipping, when 
the inclination of the plane is equal to the limiting angle of the resistance 
of the surfaces in contact ; that is, when the angle B A C is equal to the 
angle G P Q. 

EXPERIMENTS ON FRICTION, BY M. MORIN. 



SUEFACES OF CONTACT. 


WITHOUT UNGUENTS. 


UNCTUOUS 


SUEFACES. 


FRICTION OF MO- 
TION. 


FEICTION OF QUI- 
ESCENCE. 


FRICTION OF MO- 
TION. 


FRICTION OF QUI- 
ESCENCE. 


Co-effici- 
ent of fric- 
tion. 


Limiting 
angle of 
resistance. 


Co-effici- 
ent of fric- 
tion. 


Limiting 
angle of 
resistance. 


Co-effici- 
ent of fric- 
tion. 


Limiting 
angle of 
resistance. 


Co-effici- 
ent of fric- 
tion. 


Limiting 
angle of 
resistance. 


Oak upon oak, fibres parallel to the 
motion, ..... 


0.478 


250 33' 
17.53 


0.625 
0.540 


3201' 
28.23 


0.108 
0.143 


63 10' 
80 9' 


0.390 
0.314 


210 19' 
172 26' 


Oak upon oak, fibres of the moving 
body perpendicular to the motion 


0.324 


Oak upon elm, fibres parallel, . 


0.246 


13.50 


0.376 


20.37 


0.136 


7.45 






Elm upon elm, " " 










.0.140 


7.59 






Wrought iron upon oak, , 


0.619 


810 47' 


0.619 


S10 47' 










" " " wrought iron, 


0.13S 


7.52 


0.137 


7.49 


0.177 


10.3 






" " " cast " 


0.194 


10° 59' 


0.194 


10 59 






O.llS 


6.44 


" " " brass. 


0.172 


9.46 






0.160 


9.6 






Cast iron on elm, 


0.195 


11.3 






0.125 


7.S 






'• " cast iron. 


0.152 


8.39 


0.162 


9.13 


0.144 


S.12 






" " wrought iron, 










0.143 


S.9 






" " brass, , 


0.147 


8.22 






0.132 


7.82 






Brass upon cast iron. 


0.217 


12.15 






0.107 


6.T 






" " wrought iron. 


0.161 


9.9 














" " liras.<<, 


0.201 


11.22 






0.184 


7.3S 


0.164 


9.19 


Leather oxhldo, well tanned, on oak 


0.296 


16.80 














" " on oast iron, wetted 










0.229 


12.54 


2.67 


14.57 


" IjoUs on oaken drums, . 


0.27 




0.47 












" " " cast iron pulleys, 


0.23 
















Common building stones upon the 
fluino, 


0.8S to 
0.65 


20.49- 
83.2 


0.65- 
0.76 


83.2— 
86.63 











MECHANICS. 117 

From liis experiments M. Morin found, that the friction of two surfaces 
which had been considerable time in contact was not only different in its 
amount, but in its nature, from the friction of surfaces in continuous 
motion ; especially in this, that this friction of quiescence is subject to 
causes of variation and uncertainty, from which the friction of motion is 
exempt; but that the slightest jar or shock, the most imperceptible move- 
ment of the surfaces of contact, Avas sufficient to change the friction from 
the quiescent state into that wdiich accompanies motion. Hence, as every 
machine or structure of whatever kind may be considered as subject to 
such shock or imperceptible motion, all questions of construction depend- 
ing upon the state of friction should be referred to that which accompanies 
continuous motion. The friction of two surfaces outside the limits of abra- 
sion is independent of the extent of superficies, and when in motion, of 
the velocity of the motion also. 

There are three states in respect to friction into which the surfaces of 
bodies in contact may be made to pass ; one^ a state in which no unguent 
is present ; second, a state in which the surfaces are unctuous, but inti- 
mately in contact ; the third, a state in wdiich the surfaces are separated 
by an entire stratum of the interposed unguent. From experiments on 
this last class Morin deduces, "■ that with the unguents olive oil and lard 
interposed in a continuous stratum between them, surfaces of w^ood on 
metal, metal on w^ood, Avood on wood, and metal on metal, when in motion 
have all of them very nearly the same coefficient of friction, the value of 
that coefficient being in all cases included between 0.07 and 0.08, and the 
limiting angle of resistance between 4° and 4° 35^ For tallow^ as an 
unguent, the coefficient is the same as the above, except in case of metals 
upon metals, in which case the coefficient w^as found to be 0.10." 



ON THE EQUILIBErCM OF THE POLYGON OF EODS OR COEDS. 

If we take all the forces excepting those w^hich act upon the extremi- 
ties of the polygon, and find the direction of their resultant, then the two 
extreme sides of the polygon being produced will meet this direction in 
the same point. Thus, in the polygon represented loaded with the weights 
P^, P^, P^, if we find the vertical E, T passing through the centre of grav- 
ity of these weights, and produce P A and P^ B, these Avill meet R T in 
the same point T (fig. 215). 

Similarly in the funicular curve or catenary (fig. 216), if we draw tan- 
gents at the points of suspension A and B, these, being in the direction of 



118 



MECHANICS. 



the forces sustaining the curve at these points, will meet when produced 
in the vertical line G T, passing through the centre of gravity G of the 
curve. Let Gr T represent the weight of the cord A B ; draw G M and 
G N parallel to A T and 33 T ; ]^ T will represent the tension at A, and 




M T the tension at B. Such a curved line is more liable to ruptm-e, 
therefore, at the upper point of suspension, and in construction, when pos- 
sible, should be of greater dimensions. 

If a polygon of rods be reversed, that is, placed upright instead of sus- 
fended^ the position in which it will stand is that which it will assume for 
itself when loaded with the same weights, and suspended. Hence, to de- 
termine the positions in which any number of beams should be arranged 
in a polygon, so as to support one another, the timbers of a gambrel roof 
for instance; let a cord be taken, and distances be measured along it, 
equal respectively in length to the sides of the polygon ; let weights be 
attached to these, equal each to one half the sum of the weights of the two 
adjacent sides. Then the two ends of the string being held at a distance 
apart equal to the length of the base of the polygon, the form which the 
string will assume when hanging freely will be that in which the beams 
should be arranged. 

To the continual equilibrium of an u]}riglit framework it is essential 
that its joints should be stiffened. E'ow this cannot be brouglit about by 
any peculiarity in the joint itself, for the different parts of such a joint, 
being situated exceedingly near to the centre about which each rod tends 
to move, are, on the principle of the lever, readily crushed by the action 
of a force, liowcvcr slight, acting at the extremity of the rod. It is, there- 
fore, requisite that each joint should be stiffened by subsidiary framing. 
And out (►!' lli(> necessity for tliis strengthening arises the greater economy 
of the BUHpended than tlio upright polygon or framing. In the sxispeiided 
polygon or curve, tlie only precaution necessary is, that the parts should 



MECHANICS. 



119 



not tear asunder. In its upright ]30sition, tlicir flexibility as avcII as tlio 
chance of their compression^ must be guarded against. 

Tlie methods of giving rigidity to a system of rods are various. They 
all of them, however, resolve themselves directly or indirectly into the 
arrangement of the component rods in triangles. Of all simple geometri- 
cal figures, the triangle is the only one which cannot alter its form with- 
out at the same time altering the dimensions of its sides, and which cannot, 
therefore, yield, except by separating at its angles, or tearing its sides 
asunder. Hence, therefore, a triangle whose joints cannot separate, and 
whose sides are of suflacient strength, is perfectly rigid ; and this can be 
asserted of no other plane figure whatever. Thus a parallelogram may 
have sides of infinite strength, and no force may be sufficient to tear its 
joints asunder, and yet may it be made to alter its form by the action of 
the slightest force imjDressed upon it. And this is true in a greater or less 
degree of all other four-sided figures and polygons. It is for these reasons 
that in all framing, care is taken to combine all the parts, as far as pos- 
sible, in triangles ; which being once done, we know that the rigidity of 
the system may be insured by giving the requisite strength to the timbers 
and joints. 

Tlie framing of a gate presents a very simple illustration of this prin- 
ciple. The outline of the form of the gate is that of a rectangular paral- 
lelogram. If, as in the accompanying figure (fig. 21Y), the parts which 
compose it had been arranged in directions parallel to its sides only, so 




Fig. 217. 



Fig. 218. 



that the whole frame should have been composed of elementary parallelo- 
grams, each component parallelogram, and, therefore, the w^hole frame of 
the gate, would readily have altered its form. 

A bar placed diagonally across the gate remedies the evil, converting 
the elementary parts of the gate from parallelograms into triangles, and 
thus giving perfect rigidity to the frame (fig. 218). 

Further illustrations of the prmciples of framing, together with the 



120 



MECHAl^ICS. 



equilibrinm of solid bodies in contact, as in the construction of retaining 
walls and arches, will be found under the head of Architectural Drawing. 



THE MECHAl^ICAL PROPERTIES OF MATERIALS. 

To proportion properly the parts of a machine or edifice, the draughts- 
man should understand not only the kind of strain, the direction, and 
amount to which its different parts may be subjected, but also the nature 
and properties of the materials of which it may be composed, and their 
capability of resisting uninjured the required stress. Many experiments 
have been made on the strength of materials, and in the tables beneath 
will be seen great differences in select specimens of the same materials, and 
of small dimensions, showing the necessity of practical knowledge in selec- 
tion, and proper allowance against contingencies. 

The forces to which materials in constructions are subjected are com- 
j^ession^ tension^ flexure^ and torsion. 

Strength of Woods. — From the results of Experiments of Capt. T. J. 
Rodman^ of the IT. 8. Ordnance Department. 



MATERIALS, 


Specific 
gravity. 


Weight per 
cubic foot. 


Eesistance per square inch. 


Transverse 
resistance. 


' , > 








To crushing. 


To tension. 


^ = M^ 






Pounds. 


Pounds. 


Pounds. 


Pounds. 


Ash, .... 


.Slto.TS 


32 to 46 


4,500 to 8,800 


11 to 24,000 


500 to 900 


Birch, .... 


.TO 


44 


8,000 


15,000 


700 


Bass, .... 


.48 to. 50 


50 to 32 


4,600 to 5,200 


12 to 15,000 


650 


Box, .... 


.90 


56 


10,500 


23,000 




Beech, .... 


.6*7 to .73 


42 to 46 


5,800 to 6,900 


15 to 18,000 


750 


Cedar, red, 


.38 


23 


6,000 


10,000 


100 


Chestnut, 


Al to .54 


29 to 34 


5,100 to 5,600 


12 to 13,000 


350 


Cherry, .... 


.58 


36 


6,100 


12,000 


450 


Cypress, .... 


.55 


34 


8,500 


17,000 


350 


Dogwood, 


.86 


54 


7,400 


23,000 


550 


Elm, .... 


.'72to.'7'7 


45 to 48 


6,200 to 6,600 


15,000 


700 


Fir, yellow, 


.55 to. 63 


34 to 39 


7,400 to 9,200 


14 to 17,000 


400 to 600 


" red and white, . 


.46 


29 


6,600 to 7,000 


13 to 15,000 


250 to 350 


Gum, black, . 


.61 


38 


6,700 


16,000 


600 


Hickory, .... 


.82 to .95 


51 to 60 


5,500 to 9,900 


18 to 35,000 


900 


" red, . 


.72 to. 87 


45 to 54 


7,700 to 10,900 


13 to 27,000 


900 to 950 


" white, 


.90 to. 99 


56 to 62 


8,900 to 11,200 


36 to 40,000 


950 to 1,100 


Hemlock, 


.45 


28 


6,800 


16,000 


400 


Hackmatack, . 


.59 


36 






450 


Lignumvitffi, . 


1.26 


79 


9,800 


16,000 


900 


Locust, .... 


.83 


52 


9,100 


27,000 


• 800 


Mahogany, St. Domingo, . 


.76 


47 


7,400 


12,000 


660 


Maple, .... 


.68 to .73 


42 to 45 


7,700 to 8,600 


22 to 23,000 


650 


Oak, white. 


.63 to. 88 


39 to 55 


4,700 to 9,100 


12 to 21,000 


600 to 960 


" y(;llow, . 


.71 


44 


6,300 


25,000 


600 


" live, 


1. to 1.1 


62 to 69 


6,500 to 7,200 


16,000 


450 to 650 


Pino, pi tell, 


1.1 


69 


8,900 


11,000 




" white, . 


.36 to .46 


23 to 29 


5,000 to 5,800 


11 to 12,000 


350 


" yellow, . 


.53 to .67 


33 to 42 


7,800 to 8,400 


12 to 19,000 


500 to 650 


Poplar, .... 


.43 to .50 


27 to 31 


6,700 to 6,600 


8 to 15,000 


300 to 450 


Redwood, Cal., 


.39 


24 


6,100 


11,000 


250 


Spruce, .... 


.'11 


27 


5,100 to 6,800 


11 to 14,000 


850 


Teak, .... 


.96 


CO 


10,800 


31,000 


1,000 


Walnut, black, 


.53 to .65 


83 to 41 


6,800 to 7,500 


16 to 18,000 


450 to 650 



MECHANICS. 



121 



Steength of Metals. 



MATERIALS 








Specific 
gravity. 


"Weight per 
cubic foot. 


Eesistance per square inch. 




' ^ 








To crushing. 


To tension. 






Pounds. 


Pounds. 


Pounds. 


Brass, cast, .... 


8.40 


523 


10,000 


18,000 


" wire, . 














49,000 


Bronze, . 








8.70 


542 




42,000 


Copper, cast, . 








8.61 


537 




22,000 


wire, . 














61,000 


Gold, cast. 








19.26 


1,200 




20,000 


Lead, sheet, . 








11.41 


711 




3,300 


Platinum wire. 








22.0Y 


1,376 




56,000 


Silver, cast, . 








10.48 


653 




40,000 


Tin, cast, Banca, 








7.22 


450 




3,700 


" wire, 














7,000 


Zinc, cast. 








V.22 


450 




2,900 


" sheet, . 














16,000 


Iron, cast, 








7 to 7.3 


436 to 456 




13 to 25,000 


" wrought. 








7.6 to 7.8 


474 to 487 




20 to 110,000 


" cable, . 














54 to 75,000 


" wire, . 














86 to 113,000 


Steel, cast, forged \ 


3ar, 






7.78 


485 




85 to 145,000 


" soft, . 














65 to 105,000 


'* plate, . 














41,000 


Homogeneous metal, 












85 to 100,000 



Resistance to Compression. — Parts of stractui-es are usually subjected 
to a compressive force in the direction of their length, and when this 
dimension is less than a certain proportion (depending on the nature of 
the material) to its diameter or shortest side, rupture takes place from the 
absolute crushing of the parts. In the actual practice of construction, 
materials cannot with safety be subjected to any pressure approaching 
their ultimate strength. They are liable to various occasional and acci- 
dental pressures, and to others of a permanent kind, resulting from settle- 
ment, and other causes, of which no previous account can be taken, for 
which allowance must nevertheless l)e made, l^avier, from existing struc- 
tures, deduces the rule that wood and stone should not be subjected to a 
strain over one-tenth of that which breaks them, and iron to not over one- 
fourth. 

Bricks have been experimented on which have withstood a compressive 
force of 13,000 lbs. to the square inch, whilst others have failed under a 
pressure of less than 500 lbs. ; and among the buildiug stones the range 
would be even greater. The granite of Rockfort, Mass., as tested by Capt. 
Rodman, crushed under 15,300 lbs. to the inch. These experiments were 
probably made on small cubes, and are of little use to the architect ; in 
construction brick is seldom subjected to a pressure of 100 lbs. per square 
inch ; in fact, few structures have failed from an absolute crushing of either 



122 MECHANICS. 

brick or stone, but unequal settlements of tlie foundation or compression 
of the mortar in the joints may change the strain upon the material from 
a compressive to a transverse one and cause destructive cracks. The 
thickness given for walls under the head of Architectural Drawing, will 
be a sufficiently practical guide for the dimensions of such works, if 
ordinary care be used in the selection of materials and construction. 

Wood and iron, when used to resist a compressive force, are generally 
of such lengths in comparison with their sides or diameter, that rupture 
takes place partly from compression, partly from flexure. It has been 
found, that if the length of a circular post exceed eight times its diameter, 
the tendency under pressure will be to bend ; and the longer the piUar, 
the other dimensions remaining the same, the more this tendency develops 
itself. 

For cast-iron posts, the circular form is usually adopted, as being the 
strongest form for the same amount of material, the hollow pillar being in 
this respect preferable to the solid. Mr. BLodgkinson, from his experi- 
ments on cast-iron pillars, deduces the following rules : 

1st. In all long pillars of the same dimensions, the resistance to frac- 
ture by flexure is three times greater when the ends of the pillar are flat 
and firmly bedded, than when they are rounded and capable of moving. 
This shows the importap-ce of having the ends of pillars turned square, and 
of having the ends of braces square and not rounded, as has been proposed 
and adopted by some architects. 

2d. The strength of a j)illar with one end round and the other flat, is 
the arithmetical mean between that of a pillar of the same dimensions, 
with both ends rounded and both ends flat. 

2>d. A long uniform pillar, with its ends firmly fixed, whether by discs 
or otherwise, has the same power to resist breaking as a pillar of the same 
diameter and half the length, with the ends rounded. 

4:th. Some little additional strength is given to a pillar by enlarging its 
diameter at the middle j)art ; but this increase is not over one-seventh of 
the brcakino' weic^ht. 

tjth. In cast iron pillars of the same length, the strength is as tlie 3.G 
power of the diameter nearly. 

Qth. In cast iron pillars of the same diameter, the strength is inversely 
proportioned to the 1.7 power of the length. 

Tlic breaking weight of solid cylindrical cast iron pillars, with their 

^7 3.6 

ends flat and incapable of motion, is in tons -14 x-v-^, I being the length 
iu feet, d the diameter in inches. In hollow pillars the same rule ap- 



MECHANICS. 



123 



lilies, but for cP-^ we use D^-^ d^\ D being the external and d the internal 
diameter. For j)illars with ends movable and romided, one-third of the 
above formula will be the breakino- weight. 



Table I. — Diameters to the 3.G 2yower. 



Inches. 




Inches. 




Inches. 




2^ 


12.125 


5.25 


391.36 


8. 


1782.9 


2.5 


27.076 


5.5 


462.71 


8.25 


1991.7 


3. 


52.196 


5.75 


543.01 


8.5 


.2217.7 


3.25 


69.628 


6. 


632.91 


8.75 


2461.7 


3.5 


90.917 


6.25 


733.11 


9. 


2724.4 


3.75 


116.55 


6.5 


844.28 


9.5 


3309.8 


4. 


147.03 


6.75 , 


967.15 


10. 


3981.1 


4.25 


182.89 


7. 


1102.4 


10.5 


4745.5 


4.5 


224.68 


7.25 


1250.9 


11. 


5610.7 


4.75 


272.96 


7.5 


1413.3 


11.5 


6584.3 


5. 


328.32 


7.75 


1590.3 


12. 


7674.5 



Table II. — Lengths to the 1.7 power. 



Inches. 




Inches. 




Inches. 




1. 


1. 


9. 


41.900 


17. 


123.53 


2. 


3.249 


10. 


50.119 


18. 


136.13 


3. 


6.473 


11. 


58.934 


! 19. 


149.24 


4. 


10.556 


12. 


68.329 


20. 


162.84 


5. 


15.426 


13. 


78.289 


21. 


176.92 


6. 


21.031 


14. 


88.801 


22. 


191.48 


7. 


27.332 


15. 


99.851 


1 23. 


206.51 


8. 


34.297 


16. 


111.43 


i 24. 

1 


222. 



Example. — To find the breaking weight of a hollow cast iron pillar 
with square ends, whose outside diameter is 6 inches ; inside or core, 4^ ; 
and length, 16 feet. 

Then from Table I.:— Against 6. is 632.91; against 4.5, 224.68. 
632.91 — 224.68 = 408.23 ; 408.23 x 44 = 17962.12; dividing this by 



111.43, the number against 16 in Table II., we have 



17962.12 



= 162 tons 



111.43 ' 

very near of 2,240 lbs. each as the weight which would break this column. 

The above formulas apply only to long pillars ; that is, those whose 

length is at least 25 to 30 times their diameter, and the result arrived at is 



124: MECHAinCS. 

their ultimate strength ; but the permaneiit load should not exceed one- 
fourth of the breaking weight. Short cylindrical pillars may be loaded 
perfectly safe with 10 tons to the square inch of area of base. It is, of 
course, important that the columns should be straight with a square base, 
so that the direction of the strain should be through the axis. 

Tensile Strength. — Woods are seldom used to resist tensile strains, but 
when so used, the size should be very much larger than that merely re- 
quired to resist the strain, say at least ten times. Wrought iron is the 
material the most generally employed to resist a tensile force; and its 
elastic power in that capacity, or the load with which it may safely be 
trusted, is ten tons per square inch for best iron, and for ordinary iron, 
10,000 lbs. 

Transverse Strength of Materials. — The strength of a square or rectan- 
gular beam to resist lateral pressure, acting in a direction perpendicular to 
its length, is as the breadth and the square of the depth ; and inversely as 
the length, or the distance from or between points of support. Thus a 
beam twice the breadth of another, other proportions being alike, has twice 
the strength ; or twice the depth, four times the strength ; but twice the 
length, only half the strength. 

The general formula is W = — j — , in which W is the breaking weight ; 

S, a number determined by experiment on different materials (see table, p. 
120) ; h, the breadth, and d^ the depth in inches ; and Z, the length in feet. 

To find the breaking weight of a beam supported at the ends and 
loaded in the centre : Bute — Multiply the constant S, for the material from 
the above table, by the breadth and square of the depth in inches, and 
divide the product by the length in feet. 

Example. — What is the ultimate strength of a beam of white pine, 20 
feet long, 8 inches wide, and 14 inches deep ? 

S. 400 X 8 X 14^ = 627.200. '' = 31.360 lbs. 
From the above formula we can determine either the breadth, depth, or 

^W ^Tw 

length, the other quantities being known: 1) — ^^—^^ and d= /-^-^. 

To determine the depth; the weight, breadth, and length being 
known : Multiply the length in feet by the weight in pounds ; also the 
tabular number S by the breadth in inches. Divide the first product 
by the last, and the square root of the quotient will be the required 
depth. 

To find the depth of an oak timber 15 feet long and 6 inches wide, to 



MECHANICS. 125 

support a weight of 10,000 lbs. at its centre. Multiply the given weight 
by 10, and establish the depth on this basis; thus, to support 10,000 
securely and permanently , find a beam whose ultimate strength is equal to 
100,000 lbs. 

15 X 100,000 = 1,500,000. S 700 x 6 == 4,200. '^'^^I'H^^ = 357.1 

' 4,200 

1/357.1 = say 19 inches, the depth required. 

When the load is not on the middle of the beam : Divide four times 
the product of the distance of the weight in feet from each bearing, by 
the whole distance between the points of support, and the quotient is the 
equivalent length of the beam loaded in the middle. 

Suj)pose a beam 30 feet in length, with a load placed 9 feet from one 
end ; required the equivalent length. 

30 — 9 = 21. ^1 x^^ X ^ _ 25.2 feet. 

When the load is distributed over the whole length of the beam, it will 
bear double the load which it would support in the middle ; therefore, 
in calculations for the strength of a beam with distributed load, use double 
the tabular number S ; if the ends of this beam are firmly fixed, use three 
times S. K loaded at the middle with ends firmly fixed, use 14- times S. 

When a beam is fixed at one end, and the weight is placed on the 
other (fig. 218), use only one-fourth of the tabular number ; if the load is 
distributed on a like beam, use one-half of S. 

JEJx. — ^To find the depth of a white pine beam 10 inches wide, project- 
ing 5 feet from a wall, and capable of supporting with safety 2,000 lbs. 
Call the breaking weight 6 X 2.000 = 12.000 lbs. 

5 X 12.000 = 60.000. -^ X 10 = 1.000. -^^^ = 60. V&T^ 7.74 in. 

It is only necessary that the dimensions thus obtained be preserved at 
the points of greatest strain, that is, at A B (fig. 218). 
When the beam has two points of support, and the load I ^ 

is intermediate, the point of suspension of the weight is l ^ . 

the point of greatest strain, and the beam may be reduced (~^ 
towards the points of support without breaking it. Fig. 21s. 

Tlie forms of beams which afi'ord equal strength throughout are para- 
bolic (figs. 219, 220, 221), of which the axis A B and the vertex A are 
given, and the points M determined by calculations. Figs. 220, 221 are 
oftener used when the force is applied on alternate sides of A B. 

If a beam be subjected to a transverse strain, one side is compressed, 
while the other side is extended ; and therefore, where extension terminates 



126 



MECHANICS. 



and compression begins, there is a lamina or surface wMcli is neither ex- 
tended nor compressed, called the neutral surface or neutral axis. As the 
strains are proportional to the distance from this axis, the material of 
which the beam is composed should be concentrated as much as possible 




Fig. 219. 



Fig. 220. 



Fig. 221. 



at the outer surface. Acting on these principles, Mr. Hodgkinson has 
determined the most economical form for cast-iron beams or girders, of 
which the section is given (fig. 222) ; it has been found, that the strength 
of cast-iron to resist compression is about six times that to resist exten- 
sion ; the top web is therefore made only one-sixth the area of the lower 
one. The depth of the beam is generally about yV of its length, the deeper 
of course the stronger ; the thickness of the stem or the upright part 
should be from \ an inch to 1^ inches, according to the size of the beam. 
The rule for finding the ultimate strength of beams of the above section 




ezzTsi 



W7?7/977^7Z^ 



Fiff. 222. 



is : — Multiply the sectional area of the bottom flange in square inches by 
the depth of the beam in inches, and divide the product by the distance 
between the supports in feet, and 2.16 times the quotient will be tlie break- 
ing weight in tons (2240 lbs.) As has already been shown above, the sec- 
tion thus determined need only be tliat of the greatest strain, and can be 
reduced towards the points of support, cither by reducing the width of 
the flanges to a parabolic form (fig. 222), or by reducing the thickness 
of the bottom flange ; tlic reduction of the girder in depth is not in general 
as economical or convenient. 

For railway structures subject to an impulsive force, tlie up])er flange 



MECHANICS. 



127 



should be i of the lower one. For wrought iron beams, as this material 
affords less resistance to compression than tension, the top flange is generally 
made larger than the lower one in the proportion of 5 to 3. The following 
may be taken for the formula to determine the strength of solid wrought iron 
AdC 



beams : IF 



I 



in which TF is the breaking weight, A the area of 



the section, d the depth, I the distance between the suj)ports. (7 is a con- 
stant determined by experiment for each form of beam ; for the beam 
shown in section (fig. 223) (7 is found to be about 40.000 lbs. 

(2- = 1. in. X 2.75 = 2.75 sq. in. 

J = S. ^' X 0.3S0 = 3.04 " 

c = 0.42 '- X 4.3 = 1.806 " 



g^o- 



d = 9.42 



A 



\596 '' 



wmW//////////.\ 

Fig. 223. 



For wrought iron girders of large span, the box form is generally 
adopted. 

Experiments on the transverse strength of rectangular tubes ofiorought iron^ 
supported at each end, and the weight laid on the middle. 



Distance be- 
twe^ the 
supports. 


Weight of tuhes 
between the sup- 
ports. 


Breaking 
weights, exclu- 
sive of the 
weights of the 
^tubes. 


External 

depth of the 

tubes. 


External 

breadth of 

tubes. 


Thickness of the 

plates of the 

tubes. 


Feet. 




Tons. 


Inches. 


Inches. 


Inches. 


30.0 


42.62 cwt. 


57.5 


24 


16 


.525 


T.5 


72.36 lbs. 


4.454 


6 


4 


.1325 


30.0 


23.09 cwt. 


22. S4 


24 


16 


.272 


7.5 


35.53 lbs. 


1.409 


6 


4 


.065 


3.75 


9.65 lbs. 


1.1 


3 


2 


.061 


3.75 


4.84 lbs. 


.3 


3 


2 


.03 


45.0 


130.36 cwt 


114.76 


36 


24 


.75 


3.75 


9.65 lbs. 


1.1 


3 


2 


.061 


30.0 


39 cwt. 


54.3 


24 


16 


.50 



B 



C D 

Fig. 224. 



In several of these experiments, the tubes gave way by 
the metal at the top becoming wrinkled. 

In similar tubes, the strength, and consequently the 
breaking weight, is proportioned to (1.9) power of the lineal 
dimensions. 

Approxiincde formida for rivetted tubes : 



W = 



4.1D 



BD' 



h d^ i the breaking weight in tons. 



128 MECHANICS. 

In which, 

A C = D^ and a c = din inches ; h = length in feet. 

If the thickness of the metal be equal to t inches completely round 
the section, then I? = B — 2 t, and d — D — 2 z^. 

What is the breaking weight of a rectangular tube 40 feet long, depth 
2 feet 6 inches, thickness of plate \ inch, and breadth 18 inches ? 

W =^ 1 18 X 30^ — 17.5 X 29.5^ | 

^ ^ 486000 — 440167 \ = 22.96 tons. 



1600 

It is found of iron beams and tubes, that they may be safely reduced 
in strength from the middle towards the extremities in the ratio indicated 
by theory. 

It must be observed, that in the formula given for determining the 
strength of material, the force exerted is supposed to be dead weight or 
pressure, and that no consideration is paid to impulsive force, except such 
slight shocks as are incident to all structures. It is impossible to give 
rules to calculate the strength necessary to resist active forces, varying in 
intensity and frequency ; we can only give instances of practical structures 
which have been found sufficient, as mere data on which to form judg- 
ment."^ It must be remarked, that where rigidity is required, stiffness of 
beams, unlike their ultimate strength, is directly as the breadth, and the 
cuhe of their depth, and inversely as the cuhe of their length. 

Detrusion. — ^The resistance to detrusion, or the force necessary to shear 
across any material, is called into play at the joints, and in the bolts of 
framings of timber and iron. Tlie resistance of spruce to detrusion in the 
direction of its fibre, is about 600 lbs. per square inch ; of cast iron to de- 
trusion, about 73,000 lbs. per square inch ; of wrought iron, 45 to 50,000 
lbs. 

Torsion. — When two forces act in opposite du-ections upon a body, tend- 
ing to turn its extremities in different directions, or twist them, it is said to 
be subjected to torsion. Thus the main shaft of a steam-engine, at one end 
of which tlic power acts through a crank ; which at the other is transmitted 
tlirough a gear or pulley ; the resistance which the load presents on the 
one hand, and on the other, the 2)ower applied to the crank, represent two 
forces, subjecting the shaft to the action of torsion. 

When the torsion exceeds a certain limit, depending on the material 

* For tlic girders of railway bridges, they should bo of such dimensions as to bear a 
strain of two tons per foot in lengtli. 



MECHAKICS. 120 

and its form, the fibres are torn asunder, and the axles twisted off. For 
the determination of the size of axles subject to a twisting force, we de- 
duce from Weisbach the following rule, allowing a five-fold security or 
strength above the absolute breaking twist : Multiply the weight in pounds 
by the leverage in inches, and divide the product by C determined for 
difierent forms and material, and the cube. root of the product will be half 

3 l~p~ci 

the thickness of the axle, expressed by formula r = \J —7^' 

Yalue of 67 for wrought and cast iron, circidar section, 12,600 lbs. 
" " ''' " square " 15,000 " 

" ^' wood, circular '' 1,260 " 

" " '' square " 1,500 '' 

In the square section, the rule gives as the cube root of the product one- 
half the side of the square. 

Example. — ^The shaft of a turbine exerts, through a toothed wheel of 30 
inch pitch or 15 inch radius, a force of 2,500 lbs., what must be the diam- 
eter of the shaft ? 



2S00 X 15 = 37500. r = VJISs = ^m = '^'^ ^''°^''- 
l.M X 2 = 2.88, say 3 inches, diameter required. 

The length of the axle subjected to torsion does not afi'ect the actual 
amount of pressure required to produce nipture, but only the angle of tor- 
sion which precedes rupture, and therefore the space through which the 
pressure must be made to act. Tlie ultimate strength of a long shaft to re- 
sist torsion may be sufiicient, but its elasticity will be found to be too much. 

To determine the degrees of the angle of torsion of a given shaft, mul- 
tiply the load in pounds by the leverage, and also by the length of the 
shaft, between the points of the applied forces ; and divide the product by 
the fourth power of the half diameter or half square, as the case may be, 
multiplied by a constant determined by experiment, and the quotient is 

Pal 

the number of degrees in the angle, that is, ang. = ^ ^ . 

Yalue of C. 

Circular section. Square section. 

Wood, .... 3500 5800 

Cast iron, . . . 160000 280000 

Steel and wrought iron, . 280000 470000 

Example. — If the distance of the toothed wheel from the water wheel 
in former experiment be 60 inches, what is the angle of torsion ? 



130 MECHANICS. 

37.500 X 60 _ 375 x 6 ^ 375 x 6 ^ JA06 
160.000 r' 160 x 1.44* 160 x 4.28 4.28 



= 3° 3; 



An angle too considerable for practice ; it should not exceed one degree. 
To calculate tlie size of the shaft, so that the angle is \ °, the formula be- 
comes, r = V Yi — r ' ^^^ ^^ *-^® above example, 

37.500 x 60 ^ 375 x 60 ^ 375 x 6 ^ 
160.000 X i ~" 80 8 ^^-^-^^ 



V281.25 = V16.77 = 4.1. 



4.1 X 2 = 8.2, diameter of shaft. 



MECHANICAL WOEK OR EFFECT. 



To work, considered in the abstract, is to overcome, during any certain 
period of time, a continuously replaced resistance, or series of resistances. 

Mechanical work is the eifect of the simple action of a force upon a 
resistance which is directly opposed to it, and which it continuously de- 
stroys, giving motion in that direction to the point of application of the 
resistance. It follows from this definition, that the mechanical work or 
eifect of any motor is the product of two indispensable quantities or terms : 

First^ — The effort, or pressure exerted. 

Second, — The space passed through in a given time, or the velocity. 

The amount of mechanical work increases directly as the increase of 
either of these terms, and in the proportion compounded of the two when 
both increase. If, for example, the ]3ressure exerted be equal to 4 lbs., and 
the velocity one foot per second, the amount of work will be expressed by 
4x1 = 4. If the velocity be double, the work becomes 4 x 2 = 8, or 
double also ; and if, with the velocity double, or 2 feet per second, the 
pressure be doubled as well — that is, raised to 8 lbs. — the work will be 
8 X 2 = 16 lbs. ft. 

The unit of mechanical effect adopted in England and this country is 
the horse power, which is equal to 33,000 lbs. weight or pressure, raised or 
moved through a space of 1 foot in a minute of time. The corresponding 
unit employed in France is tlie kilogrammetre, which is equal to a Idlo- 
grammc raised one metre high in a second. The liorse poAver is repre- 
sented by 75 kilograuiinc'tres ; tluit is, 75 kih)g. raised 1 metre high per 
secoiuL WlicMi wt^ s])eak of small aniounls of mechanical effect, it is 
generally said that tlioy are cMpial to so many pounds raised so many feet 



MECHANICS. 131 

high in some given time, as a minute for example. Tlie time must always 
be expressed or understood. It is impossible to express or state intelligibly 
an amount of mechanical effect, without indicating all tlie three terms — 
pressure, distance, and time. 

Tlie motors generally employed in manufactures and industrial arts are 
of two kinds — living, as men and animals ; and inanimate, as water and 
steam. 

What may be termed the amount of a day's w^ork, producible by men 
and animals, is the product of the force exerted, multiplied into the distance 
or space passed over, and the time during which the action is sustained. 
There will, however, in all cases be a certain proportion of effort, in rela- 
tion to the velocity and duration which will yield the largest possible pro- 
duct or day's work for any one individual, and this product may be termed 
the maximum effect. In other words, a man will produce a greater me- 
chanical effect by exerting a certain effort at a certain velocity, than he 
will by exerting a greater effort at a less velocity, or a less effort at a 
greater velocity, and the proportion of effort and velocity which will yield 
the maximum effect is different in different individuals. 

In the manner and means in which the strength of men and animals 
is applied, there are three circumstances which demand attention : — 

1st, — The power, w^hen the strength of the animal is exerted against a 
resistance that is at rest. 

2(^. — Tlie power, when the stationary resistance is overcome, and the 
animal is in motion. And, 

^d. — ^Tlie power, when the animal has attained the highest amount of 
its speed. 

In the first case, the animal exerts not only its muscular force or 
strength, but at the same time a very considerable portion of its weight or 
gravity. Tlie power, therefore, from these causes must be the greatest 
possible. In the second case, some portion of the power of the animal is 
withdrawn to maintain its own progressive motion ; consequently the 
amount of useful labor varies with the variations of speed. In the third 
case, the powder of the animal is wholly expended in maintaining its loco- 
motion ; it therefore can carry no weight. 

Tlie followino* table exhibits the avera^-e amount of mechanical effect 
produced by men and animals in different applications ; the animal work- 
ing with a mean velocity and effort during an average day's work, thereby 
producing the maximum effect. 



132 



MECHANICS. 



Nature of the work. 


Effort exerted. 


Velocity per 
second. 


Effect per 
second. 


Duration. 


Mechanical .effect 
per day. 


Man working at a lever, as in pump- 
i^S, 


Pounds. 
10.5 


Feet. 
8.5 


37.45 


Hours. 

8 


1.078.560 


" at a crank. Length of crank 16 
to 18 inches ; height of axis of 
Bhaft, 36 to 39 inches, . 


17. 


2.4 


40.8 


8 


1.175.040 


" tread-mill at level of axis, . 


128. 


0.48 


61.44 


8 


1.769.472 


" at angle of 24° from 
the vertical, .... 


25.1 


2.25 


57.75 


8 


1.663.200 


" at a vertical capstan, . 


25 i 


1.9 


48.45 


8 


1.395.360 


Horse at a whim gin not less than 20 
feet radius, ..... 


153. 


2.9 


443.7 


8 


12.778.560 


Draught by traces, according to 
Gerstner : 












Weight. 
Man, . . . . .150 


SO. 


2.5 


75. 


8 


2.160.000 


Horse, 600 


120. 


4. 


480. 


8 


13.824.000 


Mule, 500 


100. 


3.5 


350. 


8 


10.080.000 


Ox, 600 


120. 


2.5 


300. 


8 


8.640.000 



Gerstner also gives the following formula to calculate the effect of 
change of velocity. 

P = (2— -)K 

K and c representing the effort and velocity per second, as given in the 
Table, v the assumed velocity, and P the resulting effect. 

JExample. — Suppose a horse to travel at the rate of 6 feet per second. 
What effort will he exert, and what will be the mechanical effect per 
second ? 

From the table we have <? = 4. ; K 

6 



120. ; V is assumed at 6 ; then 



p = 



-) 120 = 60. = effort. 



60 X 6 = 360 ft. lbs. effect per second. 

It is evident that this formula will not apply to extreme values of P or 
V / yet it may be considered sufficiently near for most practical purposes, 
not very different from the mean, and is illustrative of the ill effects re- 
sulting from increase of velocity. 

Water power. — Water acts as a moving power, or moves machines 
eitlier by its weight, by pressure, or by impact ; and is applied through 
various forms of wlieels. Tlie mechanical effect inherent in water is tlie 
product of its weight into the height from which it falls ; but tliere are 
many losses incurred in its application to machinery, so that only a portion 
of the mechanical effect becomes avaihible ; that is, the efficiency of any 
water wheel is represented by a certain pcr-centage of the absolute effect 
of tlie water. 



MECHANICS. 133 

Example, — ^Tlie quantity of water supplied to the mills at Lowell is 
3,596 cubic feet per second ; tlie net fall is 33 feet ; tlie absolute dynami- 
cal effect of tins water is : 

3596 X 33 X 62.33 = 7.396.576 lbs. ft. per second. 

62.33 being the weight of a cubic foot of water, at 60° Fahr. ; on an average 
it may be assumed, that a useful effect is derived equal to two-thirds of 
the total power of the water expended ; two-thirds of 7.396.576, divided 
by 550, gives 8965.5 horse power as absolutely available. 550 lbs. 1 foot 
high per second represent a horse power, being equal to 33,000 lbs. 1 foot 
high per minute. 

Steam is the elastic fluid into which water is converted by a continuous 
application of heat. It is used to produce mechanical action almost in- 
variably by means of a piston movable in a cylinder. Tlie horse power 
of a steam-engine is computed, by multiplying the area of the piston in 
square inches by the effective pressure in lbs. on each square inch of piston, 
and the product by velocity in feet through which the ^^iston moves per 
minute, dividing this last product by 33,000. 

The area of the piston is found by squaring the diameter, and multiply- 
ing the square by 0,7854. 

Example. — Let the diameter of the piston be IS inches, the effective 
23ressnre 45 lbs. per square inch, and the speed 300 feet per minute, w^hat 
will be the horse power of the engine ? 

18 X 18 X 0,7854 = 254.46 square inches, area of piston. 
254.46 X 45 X 300 



33,000 



= 104. horse power. 



To determine the effective pressure on the j^iston, recourse must be had 
to an indicator, and take the mean pressure, as sho^^m on the diagram ; the 
pressure on the boiler is readily known, but the steam in its passage to the 
cylinder is subject to various losses, as of wire-drawing, condensation, &c., 
so that frequently the pressure on the piston does not exceed two-thirds of 
that on tbe boiler. Tlie boilers of most of our stationary engines are sub- 
jected to pressures of from 50 to 75 lbs. per square inch ; the smaller en- 
gines, say less than 10 horse power, are generally worked with full steam ; 
effective pressure from 30 to 60 lbs. Larger ones are generally worked 
expansively, cutting off at from one-half to one-sixth stroke. Tlie principle 
of working steam expansively is as follows : If a cubic foot of air of the 
atmospheric density be compressed into the compass of half a cubic foot, 
its elasticity will be increased from 15 lbs. on the square inch to 30 lbs. ; 
if the volume be enlarged to two cubic feet, the pressure will be one half, 



134 



MECHANICS. 



or 71 lbs. The same law holds in all other proportions for gases and va- 
pors, provided their temperature is unchanged. 

Tims, let E (fig. 225) be a cy- 
linder, J the piston ; let the cylin- 
der be supposed to be divided in 
the direction of its length into any 
number of equal parts, say twenty, 
and let the diameter of the piston 
represent the initial pressure of the 
steam, which we call 1. If now 
the piston descend through 5 of 
the divisions, and the valve be then 
shut, the pressure at each subse- 
quent position of the piston may 
be calculated by the law above 
given, and represented as shown 
in the figure. If the squares above 
the point, when the steam was cut 
^gff, be counted, they will be found 
to amount to 50, those below to 
about 68 ; so that while, by an 
expenditure of a quarter of cylinder full of steam, we get an amount of 
power represented by 50, we get 68 without any further expenditure, by 
merely permitting expansion. Practically, for large cylinders, it may be 
stated : 

Cutting off at Saves of fuel Gains in effect 

^ stroke, 41 per cent. 70 per cent. 

J " 58 " YO X 2 = 140 per cent. 

1 " 68 " 70 X 3 = 210 " 




Mean pressure at different densities, and rate of expansion. 



Initial 
prcBBure. 


EXPANSION BY EIGHTHS. 




3 


•J 


i 


i 


i- 


i 


1 


10 


9.80G 


9.G37 


9.187 


8.405 


7.417 


5.965 


3.848 


15 


14.844 


14.456 


13.781 


12.697 


11.126 


8.947 


6.778 


20 


19.792 


19.275 


18.875 


16.980 


14.885 


11.930 


7.697 ! 


25 


24.740 


24.093 


22.903 


21.162 


18.543 


14.912 


9.621 


80 


29.CSS 


28.912 


27.502 


25.895 


22.252 


17.895 


11.M6 ! 


.r, 


84.G8C 


83.731 


88.156 


29.627 


25.961 


20.877 


13.470 


40 


89.585 


88.550 


86.750 


88.800 


29.670 


23.860 


15.895 


45 


44.588 


43.3C8 


41.848 


38.092 


83.878 


26.842 


17.819 : 


50 


49.481 


48.187 


45.987 


42.825 


87.007 


29.825 


19.243 






MECHANICS. 



135 



Water converted into steam under the pressure of the atmosphere, i. <?., 
15 pounds per square inch, expands to 1700 times its vohime ; under double 
the pressure, or 30 pounds, the volume would be one-half ; and this pro- 
portion would be strictly accurate but for the fact that the temperatures at 
which water boils in these cases are different. 

In the following table are given the total pressure of steam in pounds 
per square inch, the corresponding temperature, and the number of cubic 
inches of steam which would be produced by one cubic inch of water. 



Total pressure 

in pounds 

per square 

inch. 


Corresponding 
temperature. 


Cubic inches of | 
steam produced by 
a cubic inch of 
water. 


1 

Total pressure 

in pounds 

per square 

inch. 


Corresponding 
temperature. 


Cubic inches of 

steam produced by 

a cubic inch of 

water. 


14 


209.1 


1773 


54 


2SS.1 


516 


15 


212.S 


1669 


55 


2S9.3 


508 


20 


228.5 


1281 


56 


290.5 


500 


25 


241.0 


1044 


57 


291.7 


492 


30 


251.6 


883 


53 


292.9 


484 


35 


260.9 


757 


59 


294.2 


477 


40 


269.1 


679 


60 


295.6 


470 


45 


2T6.4 


610 


61 


296.9 


4G3 


46 


27T.S 


598 


62 


298.1 


456 


47 


279.2 


586 


63 


299.2 


449 


4S 


2S0.5 


575 


64 


300.3 


443 


49 


2S1.9 


564 


65 


301.3 


437 


50 


2S3.2 


554 


66 


302.4 


431 


51 


2S44 


544 


67 


303.4 


425 


52 


285.7 


534 


68 


304.4 


419 


53 


2S6.9 


525 


69 


305.4 


414 



It must be remarked, that in non-condensing engines, the effective pres- 
sure is the excess above the pressure of the atmosphere. 



Table showing the weights^ evaporative powers per weighty and hulk and 
character offuels^from the report ^Prof. Walter E. Johnson, 18M. 



Designation of fuel. 


Specific 
gravity. 


Weight 

per cubic 

foot. 


Water 
evaporat'd 
by one lb. 

of fuel. 


Designation of fuel. 


Specific 
gravity. 


Weight 

per cubic 

foot. 


Water 
evaporat'd 
by one lb. 

of fuel. 


BITIJMrNOUS. 




lbs. 


lbs. 


ANTnBACITE, 




lbs. 


lbs. 


Cumberland, maximum, 


1.313 


82.09 


10.7 


Peach Mountain, 


1.464 


91.5 


10.11 


" minimum 


1.337 


83.28 


9.44 


Beaver Meadow, No. 5, 


1.554 


96.9 


9.83 


Blossburgb, 


1.324 


82.73 


9.72 


Lackawana, 


1.421 


83.8 


9.79 


Newcastle, . 


1.257 


73.54 


8.66 


Beaver Meadow, No. 3, 


1.610 


100.6 


9.21 


Pictou, 


1.318 


82.83 


8.41 


Lehigh, 


1.500 


99.3 


8.93 


Pittsburgh, . 
Sydney, 
Liverpool, . 
Clover Hill, 


1.252 
1.333 
1.262 
1.2S5 


73.37 
83.66 
78.89 
80.36 


8.20 
7.99 
7.84 
7.67 


COKE. 

Natural Virginia, 
Cumberland, 


1.323 


82.70 


8.47 
8.99 


Cannelton, la. . 


1.273 


79.C4 


7.-54 


■WOOD. 








Scotch, 


1.519 


94.95 


6.95 


Dry Pine "Wood, 




21.01 


4.69 



136 MECHANICS. 

Tlie above table exhibits the ultimate effects. As a safe estimate for 
practical values, a deduction (for the coals) of jVV should be made. 

From these two tables it is easy to calculate the amount of fuel which 
must be expended to produce a given power. 

Example. — To find the consumption of water and fuel required by a 
high pressure engine, 12 inch cylinder, 4 feet stroke, the effective pressure 
on the piston being 40 lbs., and the number of double strokes 35 per niin. 

Area of piston = 12 x 12 x 0.7854 = 113.09. 
Velocity of piston = 35 x 8 = 280 ft. per min. 

Then, 113.09 x 280 x 12 = 379.982 cubic inches of steam used during 1 
minute, or 379.982 X 60 = 22.798.920 cub. in. consumption per hour. Look- 
ing in the first table against the pressure 55, that is 15, or atmosphere 
added to 40 given above, we find 508 ; dividing, therefore, 22.798.920 by 
508, we have 44880, the number of cubic inches of water used per hour ; 

-zpL-^ = 26 cubic feet nearly. 

Multiplying this by the weight of a cubic foot of water, 

26 X 62.33 = 1620.58 lbs. 

Taking the safe estimate for the anthracites of the evaporation of 8 lbs. of 
water by 1 lb. of fuel, we have, 

—^ — 202.5 lbs. the consumption of coal per hour. 

On an average of boilers, 1 square foot of grate surface is allowed for 
the consumption of 14 lbs. of coal per hour, from 15 to 25 square feet of 
heating surface, and one-sixth of a square foot of flue at the base of the 
chimney. Continuing the previous example, we have 

-f}^ ~ 1^-^ square feet of fire grate. 

14.5 X 20 = 290 square feet of heating surface. 

14.5 

— -f- = 2.42 square feet of flue. 

Tlic horse power of the above engine would be 

113,09 X 280 X 40 ^„ ^ , 

33000 = ^^'^' ^'''''' ^'''''' 

A portion of which power would be consumed in the driving of the engine 
itself, leaving about 35 horse power as effective on the first shaft. 



DKAWIXG OF MACHINERY. 



137 



DEATTIXG OF :d:ACHES^EEY. 

Havixg thus laid down tlie principles of geometrical projection, and 
tlie rules by wliicli to proportion parts, according to tlie stress to whicli 
tliev may be subjected, we now proceed to tlie practical ap2)lication ot 
tlie priiicij)les and rules in tlie di'awing of macbineiy. 



SHAFTIXG. 

Shafts are made of wood, cast and wrought iron. Fig. 226 repre- 
sents the sections of the usual forms. "Wooden shafts are mostly of an 
octagonal or polyhedral form, and are seldom used but as shafts for water- 
wheels, but are not equal 
to those of cast iron ex- 
cept in cheapness, and are 





Cast iron. 




seldom adopted when the 
latter can be readily ob- 
tained. Cast iron is used 
for the shafts of water- 
wheels, and the heavier 
kinds of mill- work, when rig. 226. 

the strain is rather transverse than torsional. The most economical form 
for cast iron shafts is the tubular, but the more usual are the feathered 
shafts, that is, with a circular or square centre, and ribs running longi- 
tudinally. AYrought iron shafts are used for the main and counter shafts 
of mills, and for heavy shafts subject to torsional or to unequal stress and 
shocks, and is by far the best material for shafts. The more usual and the 
best form is the circular. 

Shafts are termed first, second, and third movers ; the first are the first 
recipients of power, as the jack-shaft from a water-wheel, or the fly-wheel 
shaft of eno-ines : the second are the next in succession, distributing: the 



138 



DRAWING OF MACHINERY. 



power, as the main shafts of mills ; and, third, the counters or shafts trans- 
mitting the power to the machines. The strain npon a shaft may be trans- 
verse, torsional, or both. In all breast, overshot, or undershot water- 
wheels, the jack-geer may be so placed that there will be no torsional 
strain on the shaft of the wheel ; in many other shafts, no strain will be 
transmitted through the journal. In these cases, the size of the journal 
may be estimated from the transverse strain or weight to which it is sub- 
jected. Tlie following table is taken from the Practical Draughtsman^ 
calculated on this formula, D = V'w? x .1938, D being the diameter in 
inches, and w the w^eight to be sustained in lbs. 

Talle of the diameters of the journals of water-wheel and other shafts for 

heavy work. 





DIAM. OF JOURNAL IN INCHES. 




DIAM. OF JOUKNAL IN INCHES. 


Total load in pounds. 






Total load in pounds. 














Cast iron. 


Wrought iron. 




Cast iron. 


Wrought iron. 


1099.0 


2 


1.7 


100156 


9 


7.7 


2146.T 


n 


2.1 


117793 


9^ 


8.1 


8709.5 


3 


2.5 


1373SS 


10 


8.6 


5890.5 


H 


3.0 


158604 


m 


9.0 


8805.6 


4 


3.4 


182864 


11 


9.4 


12619.5 


4i 


3.8 


208950 


Hi 


9.9 


17175.5 


5 


4.3 


237296 


12 


10.3 


22858.0 


5^ 


4.7 


268012 


m 


10.7 


29676.0 


6 


5.1 


311666 


13 


11.2 


37730.0 


6i 


5.6 


338026 


13i 


11.6 


43873.0 


7 


6.0 


376993 


14 


12.0 


5S915.7 


n 


6.4 


418845 


14i 


12.5 


70353.0 


8 


6.9 


463685 


15 


12.9 


84373.0 


SI 


7.3 









Tlie length of the journal should be from once to twice the diameter. 
The size of the shaft at the point at which the load is applied may be 
determined from j)revious rules ; but for all shafts less than three feet be- 
tween bearings, the size as calculated for the journal need only be enlarged 
enough to cut the key-seat. 

ri. XIV. — Figs. 1, 2, 3, represent diiferent views of a wooden water- 
wlieel shaft. Fig. 1 shows at one end the side external elevation of the 
sluift, fumished witli its iron ferules or collars and its gudgeon; at the 
otlier end, the shiift is shown in sections, giving the ferules in section, 
l)iit nhowiug the central spindle with its feathers in an external elevation, 
(jrenerally, in h)ngitndinal sections of objects enclosing one or more pieces, 



P L A T K X I \' 



189 




DRAWING OF MACniXERY. 



139 



the innermost or central piece is not sectioned unless it has some internal 
peculiarity, the object of a section being to show and explain peculiarities, 
and being therefore unnecessary when the object is solid ; on this account, 
bolts, nuts, and solid cylindrical shafts are seldom drawn in section. Fig. 
2 is a cross or transverse section through the centre of the shaft, to show 
the outward octagonal form. Fig. 3 is an end yiew of the shaft, showing 
the iitting of the spindle B and its feathers into the end of the shaft, and 
the binding of the whole by ferules or hoops a a. Tlie spindles B, which 
are let into the ends, are cast with four feathers or wings c. Tlie tail-piece 
h is by many millwrights omitted. The ends of the beam are bored for 
the spindle, and grooved to receive the feathers ; the casting is then driven 
into its place, hooped with hot ferules, and after this hard-wood wedges are 
driven in on each side of the feathers, and iron spikes are sometimes driven 
into the end of the wood. 

Figs. 4, 5, 6, represent different views of a cast iron shaft of a water-wheel. 
Fig. 4 is an elevation of the shaft, with one half in section to show the form 
of the core ; fig. 5, an end elevation ; fig. 6, a section on the line c c across the 
centre. The body is cylindi-ical and hollow, and is cast with four feathers 
c c, disposed at right angles to each other, and of an external parabolic out- 
line. Xear the extremities of these feathers four projections are cast, for the 
attachment of the bosses of the water-wheel. These projections are made 
with facets, so as to form the corners of a circumscribing square, as shown 
in fig. 5, and they are planed to receive the keys by which they are fixed 
to the naves which are grooved to receive them. Tlie shaft is cast in one 
entire piece, and the journals are turned. 

In all line drawings, the portions of an object represented in section 
are shaded with diagonal lines, as in figs. 1, 2, 4, and 6. 

Fig. 227 represents the sec- r 
tion of a portion of a water-wheel, 
with a cast iron shaft, in use in 
this country, in which stiffness is 
given to the wheel by wooden 
trusses, and a tensional strain is 
given to the centre of the shaft. 
Tliese shafts are cast circular in 
two lencrths connected at the cen- 
tre, with circular bosses on which 
the naves of the wheel are keyed. \ 

Wlien the load upon a shaft rig. 227. 

is not central between the bearings, the size of the journals should be pro- 




140 



DRAWING OF MACHINEEY. 



portioned to the weight it will be required to support, which will be in- 
versely as their distance from the centre of pressure. 

Fig. 228 represents the fly-wheel shaft of a stationary engine. The 
parts of least diameter are the journals ; their length is 1^ times the diam- 



■r 



C 



.^ 



^ 



Fig. 22S. 




.9 "^- 



Fig. 229. 



eter ; the centre of the shaft is enlarged to receive the hub of the fly-wheel, 
and for convenience in driving the keys. Shafts of this form are mostly 
of wrought iron, the reduction being made by steps, as a convenience in 
swedging. Fig. 229 is a plan of the crank, from the wheel side. 

The torsional strain on a shaft is as the power transmitted through it. 
It is evident, power being weight multiplied by velocity, that the greater 
the velocity of the shaft, the less the strain to transmit the same amount 
of power ; and it is the modern practice to drive the shafts at high veloci- 
ties, and reduce the weight of the geering. In first movers, the strain is 
often compound ; and when the journals bear but little transverse strain, 
the determination of their size must depend entirely on their capacity to 
resist torsion. Tlie formula given in the .Practical Draughtsman for de- 
termining the proper diameter is : 



d 



=V^' 



IIP 
E 



X a 



6' being for cast iron, 1st movers, 419 ; 2d, 20G ; 3d, 100. 
" wr'ght" " " 249; " 134; " G7.0. 

Which formula is simplified and tabcllated, so that it is onl}^ necessary 
to divide tlic number or revolutions of the shaft by the horse power, and 
find llie diameter corresponding to the quotient in the table. 



DRAWING OF MACIILN'ERY. 



141 



Tcible of diameters for shaft joitmals^ calculated with reference to torsional 



strain. 



Diameter in 
inches. 


JOURNALS OF C.VST IRON' SHAFTS. 


JOURNALS 


OF WBOUGnT-IBON SHAFTS. 


First movers. 


Second movers. 


TliirJ movers. 


First movers. 


Second movers. 


Tliird movers. 


li 


124.133 


61.037 


31.408 


73.778 


39.704 


20.030 


2 


52.375 


25.750 


13.250 


31.125 


16.750 


8.450 


n 


26.S16 


13.190 


6.790 


15.872 


8.576 


4.327 


3 


15.519 


7.630 


3.922 


9.222 


4.963 


2.504 


Si 


9.773 


4.805 


2.475 


5.808 


3.123 


1.577 


4 


6.547 


3.219 


1.656 


3.S91 


2.094 


1.563 


4} 


4.598 


2.266 


1.163 


2.732 


1.475 


.742 


5 


3.352 


1.643 


.848 


1.992 


1.072 


.541 


51 


2.519 


1.239 


.637 


1.497 


.806 


.406 


6 


1.940 


.954 


.491 


1.153 


.620 


.313 


61 


1.526 


.750 


.336 


.906 


.488 


.246 


7 


1.222 


.601 


.309 


.726 


.391 


.197 


71 


1.002 


.493 


.253 


.595 


.325 


.163 


8 


.838 


.402 


.207 


487. 


.261 


.130 


SI 


.632 


.335 


.173 


.405 


.218 


.110 


9 


.575 


.282 


.145 


.341 


.184 


.093 


91 


.489 


.240 


.124 


.290 


.156 


.079 


10 


.419 


.206 


.106 


.249 


.134 


.063 


101 


.362 


.178 


.092 


.215 


.116 


.053 


11 


Q14 


1 KK 


.079 


.187 


.101 


.051 


• Oli 


.100 


111 


.275 


.135 


.069 


.163 


.089 


.044 


12 


.242 


.119 


.061 


.144 


.078 


.039 


121 


.214 


.105 


.054 


.127 


.068 


.034 


13 


.191 


.094 


.049 


.114 


.061 


.031 


131 


.170 


.084 


.043 


.101 


.054 


.027 


14 


.153 


.075 


.033 


.091 


.049 


.024 


141 


.137 


.067 


.035 


.082 


.044 


.022 


15 


.124 


.061 


.031 


.074 


.039 


.020 


1 


2 


3 


4 


5 


6 


7 



Example. — What must be the diameter of the journal of a wrought iron 
first mover, transmitting 30 horse power, and making 50 revolntions per 
minute ? 

30 = ^'^^^' 
1.667 in the table is intermediate between 1.992 and 1.49T, corresponding 
to 5 and 5|, and should be about 5f inches. 

It is the common practice to make wrought iron 2d and 3d movers of 
an uniform diameter, without reduction at the journal; the shaft is 23re- 
vented from sliding endways by collars keyed on. Tlie usual length of 
main shafts is from T to 10 feet between bearings ; and that they may run 



142 



DRAWING OF MACHINERY. 



smooth, and not spring intermediately, it is desirable that they should 
never be less than 2 inches diameter, and that the pnlleys or geers through 
which the power is transmitted to the next mover or to the machine should 
be as near as possible to the bearing. 

Fig. 230 represents a line of shafting. A is an upright shaft ; a a, 
bevel-geers ; 1) J, bearings for the shaft ; c, coupling or connection of 
the several pieces of shafting. These shafts are intended to be of wrought 
iron. No reduction is made for the journal, no bosses for pulleys or geers. 
As the power is distributed from this line of shafting, the torsional strain 
diminishes with the distance from the bevel-geers or first movers, and the 



Lt 




1 



diameter of each piece of shafting may be reduced consecutively, if neces- 
sary ; but uniformity will generally be found to be of more importance 
than a small saving of iron. The drawing given is of a scale large enough 
to order shafting by, but the dimension, should be written in. It is often 
usual in the order to the machinists merely to give the lengths of the shafts 
and diameters as thus : 



sr 8 ft. 



8 ft, -^ 



6 ft. 31- 



The X marks represent the bearings ; the joints or couplings are generally 
made near the bearings, and it is also usual to bring the pulleys as near 
the bearings as possible. It frequently happens, therefore, that the coup- 
ling and pulley are needed at the same point ; to remedy this, as the posi- 
tion of the pulley depends on the machine which it is required to drive, it 
frequently cannot be moved without considerable inconvenience or loss of 
room ; the shaft will have, therefore, to be lengthened or shortened, to 
change position of coupling ; or better, the coupling and pulley may be 
made together. 



BEARINGS OR SUPPORTS FOR TUE JOURNALS OF SHAFTS. 

Fo7' ujn'ifjht shafts. — Footstej^ or step for an aj)ri(j/it shaft. — Fig. 231 
rei)reseiits an elevation ; fig. 232, a plan of the step. It consists of a foun- 
(hition or bed-])hitc A, a box B, and a cap or socket C. The plate A is 
firmly fastened to the base on which it rests ; in the case of heavy shafts, 



DKAWING OF MACHINERY. 



143 



often to a base of granite. The box B is placed on A, the bearing snrface 
being accurately bevelled, and fitted either by planing or chipping and 



:z^ 



j: 





r 






I 








i 




; 




r^ 


B 


1 


i 
1 

i 


i 


. 




-\ 










/ '■■ '■ 


/ : ■: \ 


rL.L.^ I i i_ .\ 


f',' 






\ 'J / 






, u 




FiK. 2:32. 



filing ; I), h, h, are what are commonly called chipping-pieces, which are the 
bearing surfaces of the bottom of B. A and B are held together by two 




Fig. 233. 



screws ; the holes for these are cut oblong in the one plate at right angles 



144 



DRAWING OF MACHINEEY. 



to tliose of tlie other ; tliis admits of the movement of the box in two direc- 
tions to adjust nicely the lateral position of the shaft, after which, by 
means of the screws the two plates are clamped firmly to each other. C, 
the cup or bushing, which should be made of brass, slips into a socket in 
E. Frequently circular plates of steel are dropped into the bottom of 
this cup for the step of the shaft. The cup C, in case of its sticking to the 
shaft, w411 revolve with the shaft in the box B ; if plates are used, these 
also admit of movement in the cup. 

Eig. 233 represents the elevation of a bearing for an upright shaft, in 
which the shaft is held laterally by a box and bracket above the step. Tlie 
step B is made larger than the shaft, so as to reduce the amount of wear 
incident to a heavy shaft. The end of the shaft, and the cup containing 
oil, are shown in dotted line. Tlie bed-plate A rests on pillars, between 
which is placed a pillow-block or bearing for horizontal shaft. 

Figs. 234, 235, represent the elevation and vertical section of the sus- 
pension bearing used by Mr. Boyden for the support of the shaft of his tur- 
bine wheels. It having been found difficult to supply oil to the step of such 
wheels, it was thought preferable by him to suspend the entire weight of 
wheel and shaft, where it could be easily attended to. The shaft (see sec- 
tion) is cut into necks, which rest on corresponding projections cast in the 





Fig. 234. 



Fig. 285. 



box h ; the spaces in the box are made somewhat larger than the necks of 
tlie shaft, to admit of Babbitting, as it is termed, the box ; that is, the shaft 
l)ciug placed in its position in the box. Babbitt, or some other soft metal 
juclted, is poured in round the shaft, and in this way accurate bearing sur- 
faces arc obtained ; projections or liolos arc made in tlie box to hold the 
metal in its position. The box is suspended by lugs ^, on gimbals c, simi- 
lar to those used for nuiriners' compasses, which give a flexible bearing, 
so that the necks nuiy not bo strained by a slight sway of the shaft. The 



DRAWING OF MACHINERY. 



145 



screws e e support the gimbals, consequently tlie shaft and wheel ; by these 
screws the wheel can be raised or lowered, so as to adjust its position accu- 
rately ; beneath the box will be seen a movable collar, to adjust the lateral 
position of shafts. 

Figs. 236, 237 are the plan and elevation for the stej?^ or rather guide 
(as it bears no weight), of the foot of the shaft of these same turbines. 
The plate A is iirmly bolted to the floor of the wlieel-j)it ; the cushions, C, 
holding the shaft, are either wooden or cast iron, and admit of lateral ad- 



fl 




n 



A 



Fi?. 236. 



Fig. 237. 



justment by the three sets of set-screws. Wooden steps are often used to 
support the shafts of the smaller horizontal wheels beneath the surface of 
the water ; the fibres of the wood are placed vertically, and afford a very 
excellent bearing surface. When cast iron or steel is used for the step, it 
is usual to encase the box, and supply oil by leading a pipe, sufficiently high 
above the surface of the water, to force the oil down. 

For long upright shafts, it is very usual to suspend the upper portion 
by a suspension box, and to run the lower on a step, connecting the two 
portions by a loose sleeve or expansion coupling, to prevent the imequal 
mashing of the bevel wheels, incident to an alteration of the length of shaft 
by variations of temperature. Tlie suspension is frequently made by a 
single collar at the top of the shaft. 

When a horizontal shaft is supported from beneath, its bearing is usu- 
ally called a lyillow or ijluinber-ljloclx^ or standard'^ if suspended, the sujd- 
ports are called hangers. 

Figs. 238, 239 are the elevation and plan of a j)illow-block. It consists 

of a base plate A, the body of the block B, and the box C. The plate, as 

in the step, is bolted securely to its base ; the surface on which the block 

B rests being horizontal. A and B are connected by bolts passing through 

oblong holes, so as to adjust the position in either direction laterally. The 
10 



146 



DKAWING OF MACHINEKY. 



box or bush. C is of brass, in two parts or halves, extending through the 
block, and forming a collar by which it is retained in its place. The cap of 



vir? 



A/T~\ 



Fig. 23S 




=±t^^B=^ ;v 



\A 




Fig. 239. 



the block is retained by the screws ooo j in the figure there are two screws 
on one side and one on the other ; often four are used, two on each side, 
but most frequently but one on eacb side. 



PROJECTIONS OF A STANDAHD. 



PL XY. — The standard is simply a modification of the pillow-block, 
being employed for the support of horizontal shafts at a considerable dis- 
tance above the foundation-j^late. Fig. 1 is a front elevation ; fig. 2, a 
plan ; and fig. 3, an end elevation of a standard. Like the pillow-block, 
the plate A is fastened to the foundation itself, and the upper surface is 
placed perfectly level in both directions. On these bearing surfaces aaa 
the body of the standard rests, and can be adjusted in position horizontally, 
and then clamped by screws to the foundation-plate, or keyed at the ends. 
Fig. 4 is a i)lan of the upper part of the standard with the cover oft', show- 
ing the form of the box, with a babitted bearing surface. 

Whilst drawing the front elevation, mark oft" on figs. 3 and 4 the out- 




Fis. 5. 



h....J 


.<? 


1 
1/, 






I . 


V _- ^f- - 


V 


:::::;:L^ 


V^ ---1. - '.y..^. 


ij.K 


t 








Fi- 4. 



DRAWING OF MACHINEEY. 



147 



lines of all sucli parts as arc immediately transferable by the help of the 
square and compasses, from one figure to the other. Tlie outline e h and 
fl are arcs, whose centres lie in ^/"produced, and pass through the points 
<?, Jc andy, I. To find the j^rojection of this arc upon the plan (fig. 2), draw 
through any points m and n^ taken at j)leasure upon the arc c h (fig. 3), the 
horizontals m m, n n, and through m and 7i (fig. 1) draw^ m m and n n paral- 
lel to C D, then set off the distances o m and j9 n (fig. 3) to the corresponding 
points on the lower side of centre line M IS" (fig. 2) : thus the curve emnJc 
will be determined. By a similar method the curve c in n' will be ob- 
tained, as also the projections of all such arcs as are denoted \>-^ rc[ (fig. 3). 

To draw on fig. 3 the s t li (fig. 2), which is the line of penetration of 
two cylinders, a similar construction to the preceding may be adopted. 
But to avoid drawing too many lines on the figures, this j^rojection is con- 
structed (see fig. 5) on another part of the sheet, in which s t N represent 
the plan of the curve s t h (fig. 2), and h v^ t the elevation, as at fig. 1. 
Divide A v^ t into any number of equal parts ; let fall pei-pendiculars A h' 
y2 'y' . . . from the points of division, and horizontal and parallel lines A A, 
v'^v...\ lay off on each side from the half chords made on the semicircle, 
and we have the cuiwe h v t v s^ which may easily be transferred to its 
position in fig. 3. 

It will be observed, that one side of the elevation (fig. 1) is represented 
as broken ; this is often done in drawing, when the sides are uniform, and 
economy of space on the paper is required. 




Fig. 240. Fig. 241. 

Suspended bearings or hangers for horizontal shafts are di^-ided into 
two general classes — side hangers (figs. 2-iO, 241), and sprawl hangers 



148 DEAWING OF MACHINEEY. 

(pi. XYI., fig. 1) ; the figures will sufficiently explain the distinction. The 
side hanger is the more convenient when it is required to remove the shaft, 
and when the strain is in one direction, against the upright part ; they are 
generally used for the smaller shafts, but sprawl hangers affording a more 
firm support in both directions, are used as supports for all the heavier 
shafts. Hangers are bolted to the floor timbers, or to strips placed to sus- 
tain them, the centres of the boxes being placed accurately in line, both 
horizontally and laterally. 

PL XYI. — ^Fig. 1 represents the elevation of a sprawl hanger ; fig. 2, 
the plan looking from above, with cover of box off ; fig. 3, a section on 
the line A B, fig. 1. 

Fig. 4 represents the elevation of a bracket, or the support of a shaft 
bolted to an upright; the box is movable, and is adjusted laterally by the 
set-screws. Fig. 5 is a front elevation of the back plate cast on the post ; 
it will be seen that the holes are oblong, to admit of the vertical adjust- 
ment of the bracket. Fig. 6 is a side elevation of the box ; fig. 7, a sec- 
tion lengthways, showing aperture for grease, and the points which retain 
the babbit-metal lining in its place ; fig. 8 is a plan of the bottom half of 
the box ; fig. 9, plan of the top. 

Fig. 242 represents different views of what may be called a yoke- 
hanger. Fig. 1 is a front and fig. 2 a side elevation ; fig. 3 a plan of the 
hanger, looking up ; and fig. 4 a plan of the yoke, looking down upon it. 
A is the plate which is fastened to the beam, E is the yoke, and B the stem 
of the yoke, cut with a thread so as to admit of a vertical adjustment ; the 
box D of the shaft C is supported by two pointed set-screws passing through 
the jaws of the yoke ; this affords a very flexible bearing, and a chance for 
lateral adjustment. 

Coujplings are the connections of shafts, and are varied in their con- 
struction and proportions often according to the mere whim of the me- 
clianic making them. 

Tlie Face Coiijyling (fig. 243) is the one in most general use for the con- 
necting of wrought iron shafts ; it consists of two plates or discs with long 
strong hubs, through the centre of which holes are accurately drilled to 
fit tlie nhaft ; one-half is noAV drawn on to the shaft-, and tightly keyed ; 
tlie plates are faced square with the shaft, and the two faces are brought 
togetlier by bolts. Tlic number and size of the bolts dcj^end upon the size 
of the shaft, never less than 4 for shafts less than 3 inclies diameter, and 
more as \\\(\ (liaiiictor increases ; tlie size of the bolts varies from f to 1\ in. 
ill (lianuitcr. llie figure shows a usual proportion of parts for shafts of 
iVom 2 to 5 inches diameter; for larger than these, the ])roportion of the 
diameter of the disc to that of the shaft is too large. 



PLATE X \ i 



148 




DRAWING OF MACnDTERY. 



149 



Fig. 244 is a rigid sleeve coupling for a cast iron shaft ; it consists of a 
solid hub or ring of cast iron hooped with wrought iron ; tlie shafts are 
made with bosses, the coupling is slipped on to one of the shafts, the ends 




Fig. 242. 



of the two are then brought together ; the coupling is now slipped back 
over the joint, and firmly keyed. This is an extremely rigid connection. 





1 
1 




Fig. 243. 



Fig. 244 



Fig. 245 is a screw coupling, a very neat and excellent rigid coupling, 
for the connecting of wrought iron, more especially the lighter kinds. It 
will be observed that this coupling admits of rotation but in one direction, 



150 



DEAWINa OF MACHINEEY. 



the one tending to bring the ends of the shafts towards each other, the 
reverse motion tends to unscrew and throw them apart, and uncouple 
them. 

Fig. 246 is a clamp coupling for a square shaft. 

In many cases it occurs that rigid couplings, such as we have given, 
are objectionable ; they necessarily imply that, to run with the least strain 




Fjcr. 245. 



Fig. 246. 



possible, the bearings should be in accurate line ; any displacement involves 
the springing of the shaft, and if considerably moved, fracture of shaft or 
coupling. Wherever, then, from any cause the allignment cannot be very 
nearly accurate, some coupling that admits of lateral movement should be 
adopted. The simplest of tliese is the hox or sleeve coupling (fig. 247), 
sliding over the end of two square shafts, keyed to neither, but often held 

h 




Fi-. 248. 



in place by a pin passing through the coupling into one of the shafts. For 
round shafts, the loose sleeve coupling is a pipe or liub, generally 4 to 6 
times tlio diameter of the shaft in length, sliding on keys fixed on either 
shaft. 

Fig. 248 represents a horned coupling. The two i^arts of the coupling 
are counteiixarts of each other, each firmly keyed to its respective shaft, 
but not fastened to each other ; the horns of the one slip into the spaces 



DRAWING OF MACIIINEEY. 



151 



of tlie other ; if the faces of the horns are accurately fitted, it affords an 
excellent coupling, and is not perfectly rigid. 

It often happens that some portion of a shaft or machine is required to 
be stopped whilst the rest of the machinery continues in motion. It is 
evident that, if one half of a horned coupling be not keyed to the shaft, but 
permitted to slide lengthways on the key, — the key being fixed in the 
shaft, forming in this case what is more usually called a feather, — by slid- 
ing back the half till the horns are entirely out of the spaces of the other 
half, communication of motion will cease from one shaft to the other. 

Couplings are made on this principle, called slide or chitch couplings. 
As usually the motion is required but in one direction, the more general 
form of this coupling is given in fig. 249. A represents the half of the 
coupling that is keyed to the shaft, B the sliding half, c the handle or lever 
which commimicates the sliding movement ; the upper end of the lever 
terminates in a fork, enclosing the hub of the coupling, and fastened by 
two bolts or pins to a collar c' round the neck of the hub ; Z» is a box 
or bearing for the shaft A ; to support B the end of its shaft extends 
a slight distance into the coupling A. It w^ill be observed that the 
horns are ratchet-shaped by this form motion can be transmitted but in 
one direction ; but should it be necessary to reverse the motion, it is ne- 
cessary that the horns of the coupling be square. Shafts cannot be 




Fig. 250. Fig. 251. 

engaged with this form of coupling while the shaft is in 
rapid motion, without great shock and injury to the ma- 
chinery. To obviate this, other forms of cou]3ling are re- 
quisite ; one of these is represented (fig. 250). On the shaft 
B is fixed a drum or pulley, which is embraced by a friction 
band as tightly as may be found necessary ; this band consists of two straps 
of iron, clamped together by bolts, leaving ends projecting on either side ; 
the portion of the coupling on the shaft A is the common form of hayonet 



\J 



152 DEAWINa OF MACHINERY. 

clutch ; the part c cis fixed to the shaft, and affords a guide to the prongs 
or bayonets h h, as they slide in and out. Slipping these prongs forward, 
they are thrown into geer with the ears of the friction band ; the shaft A 
being in motion, the band slips round on its pulley till the friction becomes 
equal to the resistance, and the pulley gradually attains the motion of the 
clutch. 

But of all slide couplings to engage and disengage with the least shock, 
and at any speed, the friction cone coupling (fig. 251) is by far the best. 
It consists of an exterior and interior cone, a,h ; a is fastened to the shaft 
A, whilst h slides in the usual way on the feathery of the shaft B ; press- 
ing 1) forward, its exterior surface is brought in contact with the interior 
conical surface of a ; this should be done gradually ; the surfaces of the 
two cones slip on each other till the friction overcomes the resistance, and 
motion is transmitted comparatively gradually, and without danger to the 
machinery. It must be observed, that the longer the taper of the cones, 
the more difiicult the disengagement ; but the more blunt the cones, the 
more difficult to keep the surfaces in contact. From the table given, page 
116, it will be seen that the limiting angle of resistance for surfaces of cast 
iron upon cast iron is 8° 39', and this angle with the line of shaft will give 
a very good angle for the surfaces of the cones of this material. Wlien 
thrown into geer, the handle of the lever or shipjper is slipped into a notch, 
that it may not be thrown out by accident. 

Pulleys are used for the transmission of motion from one shaft to 
another by the means of belts ; by them every change of velocity may be 
effected. The speed of the two shafts will be to each other in the inverse 
ratio of the diameter of their pulleys. Tlius, if the driving shaft make 100 
revolutions per minute, and the driving pulley be IS inches in diameter, 
whilst the driven pulley is 12 inches, then, 

12 : 18 :: 100 : 150; 

that is, the driven shaft will make 150 revolutions per minute. Where 
there is a succession of shafts and pulleys, to find the velocity of tlie last 
driven shaft : — Multiply together all the diameters of tlie driving pulleys 
by the speed of the first shaft, and divide the product by the product of 
the diameters of all the driven pulleys. 

Pulleys arc made of cast iron and of every diameter, from 2 in. up to 20 
ft. Tlie number of arms vary according to the diameter ; for less than 8 in. 
diameter the plate pulley is preferable (fig. 252) ; that is, the rim is attached 
to the hub by a plate ; for pulleys of larger diameters, those Avith arms 
arc used, never less than 4 in number. The arms are made cither straight 



DEAWING OF MACHINERY. 



153 



(fig. 253), or curved (fig. 254). When large pulleys are cast entire, it if 
better that the arms should be curved to admit of contraction in coolinir 
for the smaller it is unimportant. 






Fig. 252. 



Fig. 253. 



Fig. 254. 



Fig. 255 represents a portion of the elevation of a pulley sufiicient to 
show the proportion of the several parts, and -Q.g, 256 a section of the same. 

Fig. 255. 




I 



1-^ 



:^^ 
^ 



M 




Fig. 256. 

The parts may be compared proportionately with the diameter of shaft ; 
thus the thickness of the hub is about | the diameter of the shaft, this pro- 
portion is also used for the hubs of couplings ; the width of the arms from 
f to full diameter ; the thickness half the width ; the thickness of the rim 
from I to i the diameter ; the length of hub the 
same as the width of face. 




aj 



..JilL 



11 1^ 



Fig. 25 T represents a faced coupling pulley, 
an expedient sometimes adopted when a joint oc- 
curs where a pulley is also required, the two are rig. 257. 
then combined ; the pulley is cast in halves — two plate pulleys, with plates 
at the side instead of central, faced and bolted together. 



154 



DRAWING OF MACHINERY. 



Wooden pulleys are commonly called drums j these are now but sel- 
dom nsed except for pulleys of very wide face. Fig. 258 represents one 
form of construction in elevation and longitudinal section. It consists of 




Fig. 258. 

two cast iron pulleys A A, or spiders, with narrow rims ; tliey are keyed 
on to the shaft at the required distance from each other, and plank or lag- 
ging is bolted on the rims to form the face of the drum ; the heads of the 
bolts are sunk beneath the surface of the lagging, and the face is turned. 

Fig. 259 represents a wooden pulley which may be termed a wooden 
plate pulley. The plate consists of sectors of inch boards firmly glued and 
nailed together, the joints of the boards being always broken. The face 
is then formed in a similar way, by nailing and gluing arcs of board one 
to another to the required width of face ; these last should be of clear 
stuff. The whole is retained on the shaft by an iron hub, cast with a plate 
on one side, and another separate plate sliding on to the hub ; the hub is 
placed in the centre of the pulley, the two plates are brought in contact 






Fi- 259. 



Fig. 2G0. 



with the Bides of the pulley, and bolted through ; the lace of the pulley is 
now turned in the lathe. A similar arrangement of hub is used for the 
hanging of grindstones. 

Cone pulleys are used to change the speed of the driven shaft. Fig. 
260 represents a cone pulley with its hangers ; on the machine there is a 



DEAWma OF MACniNERY. 



155 



similar set, but with ends reversed ; that is, the hirge end of tlie hanging 
or driving-pulley connects with the small end of the pulley on the machine. 
At this time the maximum of velocity is attained on the driven shaft ; hut 
if the belt is at the opposite end, small pulley on to a large one, the speed 
is the minimum, the speed of the shafts being in the inverse ratio of the 
diameters of their pulleys. By this arrangement speed may be varied 
within any required limit. It is not necessary that the two pulleys should 
be counterparts of each other, but only that such proportions should be 
preserved, that the belt may be tight on whatever set it is placed. 

Tlie width of the face of the pulley de^^ends upon the width of the belt 
necessary to transmit the power ; it should exceed by about half an inch 
on each side the width of the belt for the ordinary sizes. To determine the 
width of the belt, determine first as near as possible the power required to 
be transmitted. Tlie strain on the belt is determined by dividing the 
power to be transmitted by the velocity ; thus, if a belt moving at a velo- 
city of 1500 feet per min. be required to transmit 5 horse power ; that is. 



33000 X 5 == 165000 lbs. ft. : then 



165000 
1500 



= 110 lbs., the strain on the belt 



to convey the power. In addition to this strain, it must be remarked, that 
the belt is stretched on the pulleys, so that it does not sli23 while con- 
veying the power. The strain given above may be considered approxi- 
mately as the difference of tension between the two sides. Morin gives the 
following Table to determine the strain on each side of the belt. 



Portion of the 

circumference 

embraced by the 

belt. 




VALUE 


OF K. 




Xew belts on 
wooden drums. 


Ordin.ary belts 


Wet belts on iron 
pulleys. 


On -wooden drums. 


On iron pulleys. 


0.20 


l.ST 


l.SO 


1.42 


1.01 


0.30 


2.57 


2.43 


1.09 


2.05 


0.40 


3.51 


3.26 


2.02 


2.60 


0.50 


4.S1 


4.38 


2.41 


8.30 


0.60 


6.59 


5.S8 


2.87 


4.19 


O.TO 


9.00 


7.90 


3.43 


5.82 


0.80 


12.34 


10.62 


4.09 


6.75 


0.90 


16.90 


14.27 


4.87 


8.57 


1.00 
1 


23.14 


19.10 


5.81 


10.89 



Applicatio7i of the table. — Find in the table the value of K according 
to the given circumstances ; from this number subtract unit or one, and 
divide the strain on the belt to convey the power by this remainder, and 
the quotient will be the minimum tension or that on the slacJc side. Add 



156 DE AWING OF MACHINERY. 

to this quotient 10 per cent, for friction due to shafting, or other causes. 
The tension on the leading or tight belt will be the above product added 
to the strain, as given by the power required to be conveyed. 

Applying this to the example above of a strain on the belt of 110 lbs. 
with the ordinary belt embracing \ or 0.50 of the circumference, the value 
of K in the table is 2.41 ; subtract 1., = 1.41 ; 110 divided by 1.41 = 78 
lbs. ; 

78 + 10 per cent, or 78 + 7.8 = 85.8, the tension on the slack belt. 
85.8 + 110 = 195.8, the tension on the tight belt. 

Good belting of an ordinary thickness of y\ of an inch should sustain 
a strain of 50 lbs. per inch of width without risk, and without serious wear 
for a considerable time. Therefore, in the example above, the belt mov- 
ing at a velocity of 1500 feet per minute, required to transmit a power of 

five horses, should be l^ , or very nearly 4 inches in width. 

For the engaging and disengaging of a machine, that is, for putting into 
or out of motion, the arrangement of a fast-and-loose pulley is adopted as 
simpler and better than the clutches before given. It consists merely of two 
pulleys in juxtaposition on the same axis, one fast, the other loose, so that 
the belt which transmits the motion may be shifted from one to the other. 
The face of the driving pulley, that is, the one on the driving shaft, ought 
to be equal in width to that of both the fast and loose pulleys. By making 
the face of the pulleys slightly convex, the belt is prevented from slipping 
oflf, as the tendency of a belt is always to the larger diameter. 

When the belt is shifted, whilst in motion, to a new position on a drum 
or pulley, or from fast to loose pulley, or vice versa^ the lateral pressure 
must be applied on the advancing side of the belt, on the side on which 
the belt is approaching the pulley, and not on the side on which it is run- 
ning off. It is only necessary that a belt, to maintain its position, should 
have its advancing side in the plane of rotation of that section of the pulley 
on which it is required to remain, without regard to the retiring side. On 
this^)rinciple, motion may be conveyed by belts to shafts oblique to each 
other. Let A and B (fig. 261) be two shafts at right angles to each other, 

A vertical, B horizontal, so that the 
line run perpendicular to the direction 
of one axis is also perpendicular to the 

i— i" other, and let it be required to connect 

P,„ 2C1. them by pulleys and a belt, that their 

direction of motion may be as shown by the arrows and their velocities 




DRAWING OF MACHESTEEY. 



157 



as 3 of A to 2 of B. On A describe the circumference of the pulley pro- 
posed on that shaft ; to this circumference draw a tangent a h parallel to 
QYi n^ this line will be the projection of the edge of the belt as it leaves A, 
and the centre of the belt as it approaches B ; consequently, lay off the 
pulley 1) on each side of this line, and of a diameter proportional to the 
velocity required. To fix the position of the pulley on A, let fig. 262 be 
another view taken at right angles to fig. 261, and let the axis B have the 
direction of motion indicated by the arrow, then the circle of the pulley 
being described, and a tangent a' V drawn to it perpendicular to the 
axis B as before determined, the position of the pulley on the shaft A is 
likewise 



Fis. 262. 



Tlie positions of the two pnlleys are thus fixed in such a way, that the 
belt is always delivered by the pulley it is receding from, into the plane of 
rotation of the pulley towards which it is approaching. If the motion be 
reversed, the belt will run off; thus (fig. 263), if the motion of the shaft A 
is reversed, the pulley B must be placed in the position shown by the 
dotted lines. 

It is not an essential condition that the shafts slionld be at right angles 
to each other to have motion transferred by a belt. Tliey may be placed 
at any angle to each other, provided the shafts lie in parallel planes, so 
that the perpendicular drawn to one axis is perpendicular to the other. K 
otherwise, recourse must l)e had to guide-pulleys, which should be con- 
siderably convex on their face. 

Geering. — ^The term geering, in general sense, is applied to all arrange- 
ments for the transmission of power ; it is also used in a particular sense, 
as toothed geering. 

Toothed geering may be divided into two great classes — spur and hevel 
wheels. In the former, the axes of the driving and driven wdieels are 
parallel to each other ; in the latter they may be situated at any angle : if 
of eqnal size and at right angles, they are called mitre-geevs. 

Spur wheels^ strictly so called, consist of wheels of which the teeth are 
disposed at the outer periphery of the wheel (PI. XYIII.), in direction of 
radii from their centres. 



158 DRAWING OF MACHINERY. 

Internal geering^ in which the teeth are disposed in the interior peri- 
phery of the wheel, in direction of radii from their centres (plate XXYI.) 

Bach geer and pinion are employed to convert a rotatory into a recti- 
linear motion, or vice versa. In this arrangement the pinion is a spnr- 
wheel, acting on teeth placed along a straight bar (plate XXIY., fig. 1.) 

Bevel-geering^ strictly so called, consists of toothed wheels formed to 
work together in different planes, their teeth being disposed at an angle to 
the plane of their faces (plate XXII.) 

Trundle-geer or loJieel is constructed by inserting the extremities of a 
certain number of cylindrical pieces, called staves^ into equi- distant holes 
formed near the circumferences of two parallel plates. Tlie trundle or lan- 
tern is in mill- work made of wood, and is very useful when iron geers can- 
not be easily got or repaired. Tlie trundle may be used either with a spur 
wheel to transmit motion to parallel shafts, or with face or crown wheels, 
to transmit motion to shafts at right angles to each other. Face or crown 
wheels are such as have their teeth perpendicular to the plane of their faces. 
The sides of the teeth should be radial, the outer edges cornered, and in- 
serted in a single plate or disc, instead of two as the trundle. 

On the transmission of motion. — ^Tlie velocity of rotation of a driven 
wheel depends on its relative diameter to, and the velocity of the driving 
wheel with which, it is connected. Tims, if the diameter of the driven 
wheel be one-half that of the driver, then the driven wheel must make two 
revolutions for one of the driver. Tlie driver is often called a leader, the 
driven a follower. Hence, to obtain the diameters of two wheels having 
the distances apart : Divide the distances between their centres into parts, 
inversely proportional to the number of revolutions which the wheels are 
to make in the same unit of time. Thus, let A and B (fig. 264) be the 
given centres, the ratio of their velocities being respectively two and three / 
if the line joining the centres A and B be divided into 2 + 3 = 5 equal 
parts, that is, into as many equal parts as there are units in the terms of 
the given ratio, the radius of the wheel upon A will contain three of these 
parts, and the radius of the pinion on B will contain the remaining two 
parts. 

In determining the size of a pair of bevels, we are not, however, limited 
to any particular diameters as when the axes are parallel ; the wheels may 
be made of any convenient sizes, and the teeth consequently of any breadth, 
jiccording to tlic stress they are intended to bear. The question is the mode 
of dcterniiniiig \\\v. relative sizes of the pair ; and tliis resolves itself into a 
division of IIk' angle included between the two axes inversely as the ratio of 
tlioir angular velocities. Let B and C (fig. 205) be the position of the two 



DRAWING OF MACHINERY. 



UO 



given axes, and let tlieni be prolonged till tliey meet in a point A. Further, 
let it be required that C make seven revolutions while B makes four. From 





FiV. 264. 



Fis. 2G5. 



any j^oints D and E in the lines A B, A C, and perpendicular to them, 
draw D d and E ^ of lengths (from a scale of equal parts) inversely as the 
number of revolutions which the axes are severally required to make in 
the same unit of time. Thus, the angular velocity of axis B being 4 (fig. 
265), and that of the axis C being T, the line D d must be drawn — 7, and 
the line E <3 = 4. Then through d and e parallel with the axes A B and 
A C draw d c and e c till they meet in c. A straight line drawn from A 
through G will then make the required division of the angle BAG, and 
define the line of contact of the two cones, by means of which the two roll- 
ing frusta may be projected at any convenient distance from A. 

Otherwise, having determined the relative perimeters, diameters, or 
radii, of the pair, then the lines D d and E e are to each other directly as 
these quantities. 

Tlie point c may also be found more directly thus : From A towards C 
in the axis A C, set off from a scale as many equal parts (A/") as there are 
units in the number (T) expressing the velocity of that axis ; from the point 
f draw f c j)arallel to A B, and set oflf from the same scale as many parts 
(/* c) as there are units in the number (4) ex23ressing the velocity of the 
axis A B ; then a line drawn from A through c, as before, will divide the 
angle as required. 

Tlie case in which the axes are neither parallel nor intersecting admits 
of solution by means of a pair of bevels upon an intermediate axis, so 
situated as to meet the others in any convenient points. 



160 DEAWING OF MACHINERY. 

When the contiguity of tlie shafts is such as to permit of their being 
connected by a single pair, skewed bevels are sometimes employed. 

When the axes are at right angles to each other, and do not intersect, 
the wheel and screw may be employed to connect them. Tlie velocity of 
motion is in this arrangement immediately deduced from that of the screw, 
its number of threads, and the number of teeth in its geering wheel. Tlius, 
if it be required to transmit the motion of one shaft to another contiguous, 
and at right angles to it — the angular motions being as 20 to 1 ; then, if 
the screw be a single-threaded one, the wheel must liaA^e 20 teeth ; but if 
double-threaded, the number of teeth will be increased to 40, for 2 teeth 
will be passed at every revolution. If the velocities be as 2 to 1, the con- 
dition is, that the screw have half as many threads upon its barrel as there 
are teeth on the wheel ; and if 1 to 1, the wheel and screw lose their dis- 
tinctive characters : both become many-threaded screws under the form 
of wheels. Wheels of this sort may often be applied with peculiar advan- 
tage, especially in light geering ; and when so applied, it is not essentially 
necessary that the axes be at right angles to each other auy more than it 
is in bevel-geer. 

If the screw have few threads compared with the number of teeth of 
the wheel, it must always assume the position of driver on account of 
the obliquity of the thread to the axis ; and in this respect its action is 
analogous to that of a travelling rack, moving endwise one tooth, whilst 
tlie screw makes one revolution on its axis. 

On tlieintcli of wheels. — Tlie primary object aimed at in the construc- 
tion of tootlied-geer is the uniform transmission of the power, supposing 
that to be constant and equal. This implies that the one wheel ought to 
conduct the other, as if they simply touched in the plane, passing through 
both their centres. Tliis plane is denoted by the line A B in iig. 264. 

When this line — which is usually denominated the line of centres — is 
divided into two parts, A c and B ^, proportional to the number of teeth 
formed upon the perimeters of the pinion and wheel, these two parts 
are jpro^portional or primitive radii of the pair ; and a circle being de- 
scribed from each centre passing through the common point c, limits what 
is called the pitch line or circle ; that is, a circle described from the centre 
A, and another from the centre B, through the same point, are called, the 
first, iha jntch circle or jjitch line of the pinion, and the other of the wheel. 
Tliey arc also sometimes called i\\Q primitive and proportional circles. If 
the pitch circh; be dlvlikid into as many equal parts as there are teeth 
to be given to the wheel, the length of one of these parts is termed the 
jntch of the teeth. Ojie of those arcs comprehends a complete tooth and 



DRAWING OF MACHINERY 



IGl 



sj^ace^ meaning by sjpace tlie liollow opening between two contiguous 
teetli. In bevel and conical wheels, the pitch circle is the base of the 
frustum. 

Rules. — I. To find the pitch of the teeth of a wheel, the diameter and 
number of teeth being given, divide the diameter D (in inches) by the 
number of teeth ^N", and multiply the quo- 
tient by 3.1410 : the product is the pitch 
in inches or parts of an inch. 

II. To find the diameter of a wheel, 
the number of teeth and pitch being 
given, divide the pitch by 3.1416, and 
multiply the quotient by the number of 
teeth. 

III. To find the number of teeth, the 
diameter and pitch being given, divide 
3.1416 by the pitch, and multiply the re- 
sult by the diameter in inches. 

In ordinary geering, the pitches most 
commonly in use range from 1 inch to 4 
inches, increasing up to two inches by 
eighths^ and beyond by fourths of an 
inch. Below inch the pitches decrease 
by eighths down to \ inch. 

The rules given above may be greatly 
simplified by the use of the aim.exed 
table, which will be found very conven- 
ient when the diameter D is to be deter- 
mined, the pitch P and number of teeth 
K being given ; and conversely, when 
the diameter and pitch are given, to find 
the number of teeth. 

Ex. 1. — Given a wheel of 88 teeth, 
2-|-inch pitch, to find the diameter of the pitch circle. Here the tabular 
number in the second column answering to the given j)itch is .Y95S, which 
multiplied by 88 gives 70.03 for the diameter required. 

2. Given a wheel 33 inches diameter. If -inch pitch, to find the number 

of teeth. Tlie corresponding factor is 1.Y952, which multiplied by 33, 

gives 59.242 for the number of teeth, that is, 591 teeth nearly. ]^ow 59 

would here be the nearest whole number, but as a wheel of 60 teeth may 

11 





P 

D=-- X N. 


N = — X D 


Pitch in inchfis 
and parts of an 


TT 


P 


Rule. — To find 
the diftm. in inches, 


Rule. — To findi 
the number of teeth, 




multiply the number 
of teeth by the tabu- 


multiply the given 
diameter in inches 




lar number answer- 
ing to the given 
pitch. 


by the tabular num- 
ber answering to the 
given pitch. 


Values of P. 


P 

Values of — 

TT 


Values of — 
P 


6 


1.9095 


.5236 


5 


1.5915 


.6283 


^ 


1.4270 


.6981 


4 


1.2732 


.7854 


3i 


1.1141 


.8976 


3 


.9547 


1.0472 


n 


.8754 


1.1333 


Oi 


.7958 


1.2566 


2i 


.7135 


1.3963 


2 


.6366 


1.5708 


n 


.5937 


1.6755 


If 


.5570 


1.7952 


If 


.5141 


1.9264 


u 


.4774 


2.0944 


If 


.4377 


2.2S4S 


u 


.3979 


2.5132 


H 


.3563 


2.7926 


1 


.3183 


3.1416 


i 


.2785 


3.5904 


* 


.2387 


4.18SS. 


1 


.1989 


5.0266- 


i 


.1592 


6.2832 


f 


.1194 


8.3n6 


1 


.0796 


12.5664 



162 



DRAWING OF MACHINERY. 



be preferred for convenience of calculation of speeds, we may adopt that 
number, and find the diameter corresponding. The factor in the second 
column answering to If pitch is .557, and this multiplied by 60 gives 33.4 
inches as the diameter which the wheel ought to have. 

Another mode of sizing wheels in relation to their pitches, diameters, 
and number of teeth, is adopted in some machine shops, by dividing the 
diameter of the pitch circle into as many equal parts as there are teeth to 
be given to the wheel. To illustrate this by an arithmetical example, let 
it be assumed that a wheel of 20 inches diameter is required to have 40 
teeth ; then the diametral pitch, 

20 1 



40 



— — ■= \ inch ; 



m 



that is, the diameter being divided into equal parts corresponding in num- 
ber to the number of teeth in the circumference of the wheel, the length 
of each of these parts is \ an inch, consequently m = 2 ; and according to 
the phraseology of the workshop, the wheel is said to be one of two joitch. 
In this mode of sizing wheels, a few determined values are given to 
m, as 20, 16, 14, 12, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, which includes a variety 
of pitches from | inch up to 3 inches, according to the following table, 
which shows the value of the circular pitches corresponding to the assigned 
values of m. 



Values of m. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


12 


14 


16 20 


Corresponding circular pitch in deci- \ 
mals of an inch, f 


3.142 


1.571 


1.047 


.785 


.628 


.524 


.449 


.393 


,349 


314 


262 


.224 


1961.157 

1 



As it is convenient to express all the dimensions in terms of the same 
unit, and the pitch being an appropriate quantity, it is nearly universally 




adopted as tlio tci-m of comparison. Tlie following are the proportions 
adopted by dilfcrent workshops, some preferring one and some the other 
(fig. 2Cfi). 



1 pitch, 


or = 15 parts, 


_3_ ii 

1 


" = 5i " 


G a 

1 


u ^ 11 u 


r'o 


i^ = I u 


r o" 


u ^ 12 " 


r\ " 


- = 7 '' 


i^- " 


'^ - 8 " 



DEAWING OF MACHINERY. 1G3 

A C = Pitch of teeth, 

a G = Depth to pitch line, P P, 
Aa + ac = Working depth of tooth, 
Co — A a = Bottom clearance, 

C a = AYhole depth to root, 

C h = Tliickness of tooth. 

Ah = Width of space. 

In practice, these proportions are often laid off in lines for the conveni- 
ence of the workmen in the pattern shop, so that for any given pitch the 
other dimensions may at once be determined by means of the compasses. 
In figs. 2GT and 268, two diagrams of that sort are given. Fig. 267 con- 
tains the proportions last enumerated, in which the pitch is snj^posed to be 
divided into 15 equal parts ; and fig. 268 is constructed nearly according 
to the proportions first given, but embraces the recognised principle, that 
the relative amount of clearance ought to vary inversely as the pitch, 
wheels of small pitch requiring more clearance relatively than those in 
Avhich the pitch is greater. 

The construction of these scales is very simple. Thus, in fig. 267, let 
A B be divided into 15 equal parts, and draw B C perpendicular to it ; and 
again divide B C into a determinate number of parts from B, actual meas- 
ures of the pitches for which the scale is intended to be used ; that is, B a 
= 1 inch ; B J = 1 inch ; B c = 2 inches, and so on, and join a and A, 
h and A, c and A, and so on. To complete the scale, draw 15 parallels to 
B C from the points numbered in the line A B, and also the two parallels 
T and IT equidistant from the parallels on each side of them. 

Tlie scale is thus ready for use. To get from it the several proportions 
for a given pitch, say of three inches = B d, let the compasses be extended 
from the intersection of the parallel marked T, with the line A B, to the 
point where it intersects the line A d; this will be the part of the tooth 
from the pitch line to the point, and equivalent to 5|- j)arts of the pitch, 
(viz., of B d) ; similarly, the compasses being extended from the intersec- 
tion of the parallel U with the line A B to its point of intersection of the 
line A d, will give the part of the length of the tooth from the pitch line 
to the root, and equivalent to 6| parts of the pitch. For the whole length 
of the tooth (if wanted in one measurement), set the compasses to the point 
where the parallel marked 12 meets the line A B, and extend to its point 
of intersection of the line A d at ,?, the length is 12 parts of the pitch B d; 
the working depth is in like manner found from the parallel marked 11, 
the thickness from that marked 7, and the width of space from that 
marked 8. 



164 



DKAWING OF MACHINERY. 



The proportions for any other given pitch comprised in the scale are 
found in precisely the same way ; and if the scale be well constructed, they 
may be measured off with the utmost accuracy and readiness. To save 
confusion, it is, however, better in practice to insert in the diagram only 



Projyortion scales for geering. 



d Fig. 267. 




8:s 8:4 ein 7.0 7:2 K,d »jy c.'o &:(i 6:2 «:a t\i AjO a:6 •i:i s;a »;& 20 



is 8 isai 



rt2 111 uit jn 18 ]7 Mi 13 11 . ]» lu 11 10 8 1 



4 3 It J. 



11 IJ lit XL 10 



^±H]: 



those parallels, namely, T, U, 12, 11, 8, 7, whicli are required; the others 
are not requisite, and l)y inattention may lead to error. 

The dc8crij)ti<)n of the scale as here given supposes that the lateral 



DRAWING OF MACHINERY. 165 

clearance is constantly l-15tli of tlie pitch ; but as it is commonly desirable 
that this should vary slightly with the pitch, relatively increasing as the 
pitch decreases, two other lines, m n and j9 (/^ ^^i^^'c been introduced into 
the scale, to enable such modification to be adopted should it be required. 
Tliese lines are drawn at such angles as to give a clearance at 6 inches 
pitch of l-18th, which is increased at |-inch pitch to 1-lOth. From these 
lines the thickness and space are to be taken, instead of using the lines 
marked T and 8, setting the compasses in the points of intersection with 
the pitch lines, and extending perpendicularly to the line A B ; in other 
words, the shortest distance from the point of intersection with the pitch 
line to the line A B is the required measure of the space when the 
line J9 (^ is taken, and of the thickness of tooth when the line vi n is 
taken. 

Fig. 268 is more comj)lete than the one described ; the principle of its 
construction is in effect the same, but its use is more extended, the diam- 
eter of the wheel being found from it simultaneously with the length and 
thickness of tooth, width of space, and clearances. The scale is adapted to 
wheels of all the pitches, from \ inch up to 3 inches. The mode of con- 
struction is this : having drawn the line A D of any convenient length, 
raise the perpendicular C B to it, also of any convenient length. On the 
line A D lay off the greatest pitch of the scale from C to A ; then from 
C towards D lay off seven times the pitch once or twice^ according to the 
sizes of wheels to which the scale is intended to be applied. In the scale 
given, double of seven times the pitch is laid off, namely, 42 inches ; then 
each of these great divisions being subdivided into 11 equal parts, one of 
these parts will be equal to four teeth upon the radius of the wheel, so that 
the whole line C D w^ill be divided into 88 radial pitches. Next on the 
line C B set off the pitches which may be required in the scale, and 
through these points draw the 24 parallels to A D, terminating in the lines 
A B and D B. Then each parallel measured from the line B C to its 
point of termination in B D is the radius of a wheel of 88 teeth of the par- 
ticular pitch marked against it on the line A B. They also express the 
radii of wheels having less than 88 teeth when measured only to the 
corresponding point in the line joining B, and the divisional on C D, 
against w^hich the number of teeth is marked. Thus, the radius of a 
wheel of 52 teeth and l|-inch j)itch is r « = 15 7-16tli inches very nearly. 
(The true answer by the table, page 161, is 30.8724 -^ 2 = 15.4362 
inches). 

The scale may also be used when the number of teeth exceeds 88 ; for 
example, to find the radius of a wheel having 100 teeth. Tlius, having 



166 DRAWING OF MACHmEEY. 

found the radius answering to 88 teeth, upon the same parallel take off the 
measure answering to the difference 100 — 88 = 12 teeth ; and the two 
measures together will be the radius required. 

To adapt the scale to odd numbers of teeth, the first division on the 
right of C is divided into single radial pitches, so that the radius of any 
wheel may be measured off without having recourse to calculation of any 
kind. Thus, for examj)le, if the wheel is intended to contain 50 teeth, the 
parallel, answering to the particular pitch, comprehended between 62 B 
and 2 B, will give the radius required, that is, a radius answering to 
52 — 2 = 50 teeth ; and any other number of teeth, when not marked 
against the base, may be found in the same way, observing that it is more 
convenient to subtract than to add in this use of the scale. 

For the proportions of the teeth, set o& C a = 7-tenths of the pitch, 
then will A a = 3-tenths of the pitch, which corresponds to the depth from 
the point of the tooth to the pitch line. Again, set off C J = l-ffteenths 
of the 3-inch pitch, and ^-elevenths on the parallel against the 1-inch pitch ; 
this will be the thickness of the tooth, allowing from a fifteenth for clear- 
ance on the largest pitch to a tenth on those from |-inch and under ; and 
A h will be the width of space, including the clearance. Lines being 
drawn from those points to B complete the diagram, which wall be foimd 
to contain all the proportions enumerated in the preceding table. 

To use the scale, lay off the addendum of the tooth ; that is, the length 
beyond the pitch line, equal to A ^ = fV pitch, and the same length 
marked off within the pitch line will give the whole working depth of the 
tooth, namely, 6-lOths pitch. Then with the measure Q a =■ j\ pitch in 
the compasses, mark off the whole length of the tooth, and this will allow 
1-lOth at bottom for clearance. Again, set off the thickness of tooth = 
C h, and the space = Ah, which will contain the clearance for the par- 
ticular pitch, varying from l-15th to fully 1-lOth on the small pitches. It 
is hardly necessary to observe, that these measurements must be taken 
upon the parallel corresponding to the particular pitcli under considera- 
tion at the time. 

Tlic amount of bottom clearance is here presumed to be uniformly 
1-lOtli of the pitch ; but if it be thought advisable to make this vary, as in 
tlio case of the lateral clearance, it will then be necessary to insert a third 
line c B in the scale, and so related to (^ B tluit tlie space a c shall be 
tlirougliout ecpial to tlic depth of tooth from the pitch circle to the root, 
and giving any l)<)tl()ni clearance that may be desired. 

Ill rchitloii to tlio siiviigtli of wheels, M. Morin gives it as a rule, that 
when the velocity of the pitch circle does not exceed five feet per second, 



DEAWING OF MACHINERY. 



167 



tlie breadth of the tooth measured parallel to the axis ought to be equal 
to four times the thickness ; but when the velocity is higher, the breadth 
ought to be equal to five thicknesses, the teeth being constantly greased. 
It* the teeth be constantly wet, he recommends the breadth to be made 
equal to six thicknesses at all velocities. 

With respect to the thickness of the tooth, it is j^lain that it must be 
dependent on the pressure which the tooth is required to sustain, and upon 
the nature of the material of which it is formed. We subjoin the follow- 
ing: table for calculatino; the strength of cast iron wheels. 

For teeth of wood add 50 per cent, to 
the thickness as given by Table. 

When cast iron and wooden teeth 
work together, their action upon each 
other tends, in consequence of the elas- 
ticity of the wood, to maintain a more 
uniform distribution of the strain, and 
being at first commonly more accurately 
dressed, to prevent abrasion of wood by 
the iron, they work with much less fric- 
tion, are less liable to shocks, and nearly 
exempt from accident by hard particles 
coming between the teeth. 

The best practice, when a mortise and 
iron wdieel are to work together, is to 
make both of the same pitch, and in the first instance, of the same thick- 
ness of tooth — ^the pitch being of course calculated for the wooden teeth ; 
afterwards, to dress down the teeth of the iron wheel by the chipping-tool, 
or the geer-cutting machine and file, to the exact form and thickness, as 
given by the Table ; that is, to a thickness in relation to the thickness of 
the wooden teeth, which shall have the ratio of 25 to 38. The following 
table may be useful in determining the relation of the dimensions of the 
teeth of wheels of the given pitches, and the power which they are capable 
of transmitting safely at the various speeds named. 

To find the power which a wheel is capable of transmitting for other 
velocities than those in the table : — For 6 feet per second, double the result 
given for 3 feet ; for 8 feet, double the result at 4 feet, and so on ; and for 
lower velocities than those given, divide the tabular number by the ratio 
which they bear to those enumerated. Thus, for 2^ feet velocity, take 
half the result at 5 feet, and so of other velocities. 



lllii IcLUXKD 


x\jx \^cixu 


LI.J.ClLi.11" UXX*^ i3Li» 


Stress in lbs. 

at the pitch 

circle. 


Thicknew of 
teeth. 


Actual pitches to which 

the wheels would be 

made. 


lbs. 


inches. 


inches. 


400 


0.50 


H to u 


SCO 


0.71 


H " n 


1200 


0.S7 


It " 2 


1600 


1.00 


2 " 2i 


2000 


1.12 


2i " 2i 


2400 


1.22 


2i " 2i 


2S00 


1.32 


2i " 2J 


3200 


1.41 


2i " 3 


3600 


1.50 


3i " 3J 


4000 


1.5S 


3i " 3| 


4400 


1.66 


3f " 3^ 


4S00 


1.73 


31 " 3i 


5200 


l.SO 


3i " 3i 


5600 


1.87 


3} " 4 


6000 


1.94 


4 « 41 



168 



DRAWING OF MACHINERY. 



When a wlieel and pinion, which differ veiy much in size, work 
together, the teeth of the latter, on account of their unequal thickness, are 
capable of sustaining much less pressure than the teeth of the wheel : they 
are in effect, if not in fact, much reduced in thickness ; and in applying 





Thickness 
of teeth. 


Length 
of teeth. 


Least 


Velocity of the 


wheel at the pitch circle. 


Pitch. 


breadth of 
teeth. 


Three feet 


Four feet 


Five feet 


Seven feet 


Eleven feet 










per second. 


per second. 


per second. 


per second. 


per second. 


Inclies. 


Inches. 


Inches. 


Inches. 


H. P. 


H. p. 


H. p. 


H. P. 


H. p. 


6 


2.9 


4.2 


8.4 


43.2 


57.6 


72. 


100.2 


158.4 


5^ 


2.6 


3.85 


7.7 


36.3 


48.4 


60.5 


84.7 


133.1 


4 


1.9 


2.8 


5.6 


19. 


25.5 


32. 


45. 


70.5 


Si 


1.6 


2.45 


4.9 


14.75 


19.5 


24.5 


34.25 


54. 


3 


1.4 


2.1 


4,2 


11. 


14.5 


18. 


25. 


39.5 


n 


1.2 


1.75 


3.5 


7.5 


10. 


12.5 


17.5 


27.5 


2 


0.95 


1.4 


2.8 


4.75 


6.25 


8. 


11. 


17.25 


If 


0.83 


1.225 


2.45 


3.5 


5. 


6.25 


8.5 


13.5 


n 


o.n 


1.05 


2.1 


2.75 


3.5 


4.5 


6.25 


10. 


n 


0.59 


0.875 


1.75 


2. 


2.5 


3.125 


4.2 


6.8 


n 


0.53 


0.7S75 


1.575 


1.5 


2.25 


2.5 


3.5 


5.5 


1 


0.4S 


0.7 


1.4 


1.2 


1.6 


2. 


2.S 


4.4 


1 


0.41 


0.6125 


1.225 


1. 


1.4 


1.75 


2.5 


3.8 


i 


0.36 


0.525 


1.05 


.7 


.9 


1.125 


1.5 


2.5 


i 


0.33 


0.4375 


0.875 


.5 


.625 


.75 


1. 


1.7 


i 


0.24 


0.35 


0.7 


.3 


.4 


.5 


.7 


1.1 


f 


0.18 


0.2625 


0.525 


.2 


.25 


.3 


.4 


.6 


i 


0.12 


0.175 


0.35 


.075 


.1 


.125 


.175 


.275 



rules to the calculation of the strength of wheels, the difference of size of 
the pair ought not to be overlooked, unless, as is indeed very common in 
practice, the deficiency of strength be made up to the pinion by a flange 
cast on one or both sides of the rim, of the same depth as the teeth, and 
binding these together like the staves of a trundle. In this case the 
pinion is commonly the stronger wheel of the pair. 

In the construction of wheels, the problem which presents itself relative 
to the shape of the teeth is this, that the surfaces of mutual contact shall 
be so formed, that the wheels shall be made to turn by the intervention of 
the teeth, precisely as they would by the friction of their circumferences. 

Fundamental jpTinci])le. — In order that two circles A and B (fig. 269) 
may be made to revolve by the contact of the surfaces of the curves in iii 
and n n of their teeth precisely as they would by the friction of their cir- 
cumferoucoR, it is necessary and sufhcient that a line drawn from the point 
of contact t of tlie teeth to the point of contact c of the circumferences 
(pitch circles), shoiihl, in every position of the point t^ be pei'pendicular to 
the surfaces of contact at that point ; tliat is, in the language of matlie- 
maficians, that the straight lino bo a normal to both the curves ni ni and 



DRAWING OF MACniNERY. 



169 



01 n. The principle liere announced exhibits a special application of one 
particular property of that curve known to mathematicians as the epicy- 
cloid (see page 76). 





Fig. 269. 



Fig. 270. 



Of eficycloidal teeth. — The simplest illustration of the action of epicy- 
cloidal teeth is when they are employed to drive a trundle, as represented 
in fig. 270. Let it be assumed that the staves of the trundle have no sen- 
sible thickness ; that the distance of their centres apart, that is their ^^^c^, 
and also their distance from the centre of the trundle, that is their pitch 
circle, are known. Tlie pitch circles of the trundle and wheel being then 
drawn from their respective centres B and A, set off the pitches upon these 
circumferences, corresponding to the number of teeth in the wheel and 
number of staves in the trundle ; let five j)iiis a h c, &c., be fixed into the 
pitch circle of the trundle to represent the staves, and let a series of epicy- 
cloidal arcs be traced with a describing circle, equal in diameter to the 
radius of the pitch circle of the trundle, and meeting in the points h I m n^ 
&c., alternately from right and left. If, now, motion be given to the wheel 
in the direction of the arrow, then the curved face m r will press against 
the pin ^, and move it in the same direction ; but as the motion continues, 
the pin will slide upwards till it reaches m, when the tooth and pin will 
quit contact. Before this happens, the next pin a will have come into 
contact with the face co I of the next tooth, which repeating the same 
action, will bring the succeeding pair into contact ; and so on continually. 

To allow of the required thickness of staves, it is sufificient to diminish 
the size of the teeth of the wheel by a quantity equal to the radius of the 
staves (sometimes increased by a certain fraction of the pitch for clear- 



170 



DRAWING OF MACHINEEY. 



S-\--- 



ance), by drawing within tlie primary epicycloids at the required distance 
another series of curves ]3arallel to these. In practice, a portion mnst be 
cut from the points of the teeth, and also a space mnst be cut out within 
the pitch circle of the driver, to allow the staves to pass ; but no particular 
form is requisite, the condition to be attended to is simply to allow of suffi- 
cient space for the staves to pass without contact. 

The action of a wheel and trundle being understood, it is easy to com- 
prehend that of the teeth of a pair of wheels of the ordinary construction. 
Let A and B of fig. 271 be respectively the centres oi a wheel and pinion 

of which the teeth are intended to be 
of the epicycloidal form, and A c and 
B G their primitive radii. To lay oflf 
the teeth of this pair, having deter- 
mined the pitch and number of teeth 
in the wheel and pinion, let the pitch 
lines be divided into as many equal 
parts, setting out from the point of 
contact c, as there are teeth in them 
respectively. Let the thickness of 
the teeth be next set ofif, taking c a 
for the thickness of a tooth of the 
wheel, and c h for that of a tooth of 
the pinion. Upon the radii A c and 
B c as diameters describe two circles, 
having also their j^oint of contact at 
c and their centres at X and Y. I^ow 
let the circle Y be made to roll upon the pitch line of the wheel, and a 
point in its circumference at c will describe the epicycloidal arc c m^ and 
this curve determines the form of the point of the tooth of the wheel. Li 
the same way describe the epicycloidal arc c n, by making the circle X to 
roll upon the pitch circle of the pinion, and this curve will determine the 
form of the part of the tooth of the pinion beyond the ^^itch line. 

Tlie curve c ?7^ of the tooth of the wheel is constantly in contact with the 
radius B c, and its point of contact is at the same time situated in the circum- 
ference of the circle X ; the contact will therefore cease when the extrem- 
ity 771 of the tooth l)ccoincs the point of contact ; and this occurs when the 
point m has arrived at tlie circumference of Y. If, then, an arc of a circle be 
described from tlie centre A with the radius A ?/?, its point/of intersection 
with the circumference of Y is that at which the tooth ought to cease to act, 




-J..-^- 



Fig. 2T1. 



DRAWING OF MACHINERY. 171 

to secure uniformity of motion, and at the same instant tliat another new 
tooth advances into geer. The determination of the point/* limits also the 
useful length of the flank ; for if from B as a centre, with radius By, an 
arc be described, the part cj of the radius c B is that which is in contact 
with the tooth till it arrives at the position B/*, and consequentlj this is 
the useful length of the flank of the pinion. In the same manner it may- 
be determined, that the useful length of the flank of the tooth of the wheel 
G a\^ the portion c g. 

To find the form of the portions of the teeth within their respective 
pitch circles, it is usually considered enough that sufficient sj^ace for play 
be allowed whilst the tooth remains strong enough for its work. As every 
tooth moves between two flanks, and touches only one of them, the space 
may be bounded literally by radial lines prolonged towards the centre of 
the wheel. Tlie bottom of tlie space being sufficiently removed from the 
pitch circumference to allow the tops of the teeth of the pinion to pass 
round without touching them, may be described by arcs of a circle drawn 
from the centre. In practice, and especially when the teeth of the wheels 
are small, it is not usually considered necessary to apply strictly the form 
of the epicycloid to find cmwes of the teeth, but to define them approxi- 
mately. 

The forms of the teeth are also occasionally described by arcs, of which 
the radii are equal to the pitch, and with the centres taken upoii the pitch 
lines. Wlien the diameters of the wheels are not very unequal, and the 
teeth not thick, in many cases the ciuwes of the teeth are described by arcs 
drawn from the centres of the adjacent teeth upon the pitch line. This 
gives a radius equal to the pitch, ^lus half the thickness of a tooth. In 
like manner, the sides of the teeth also are sometimes described by arcs 
from the centres of the adjacent teeth, giving a radius equal to the pitch, 
minus half the thickness of a tooth. This form of the tooth may be defec- 
tive, in the case of a very small pinion having to transmit great pressure, 
as the extremities of the teeth may be too much reduced. In this case, 
the curves or faces of the teeth may be described with radii equal to three- 
fourths of the pitch ; and if this be not sufficient curvature, radii equal to 
some smaller fraction of the pitch may be used. When, on the contrary, 
the pitch is large, and the pressure comparatively small, the teeth may be 
too short ; this will be remedied by employing arcs, of which the radius is 
one and a half or twice the pitch. In practice, the ordinary mode in 
which epicycloidal teeth are set out is by the mechanical method of form- 
ing the pattern teeth by temjplets (figs. 272, 273, 274, 275). 

Having determined the pitch of the teeth and the radius of the pitch 



172 



DRAWING OF MACHINEEY. 



circle^ describe on a thin slip of wood — say I in. thick — an arc of the pitch 
circle, and on another similar slip an arc of a circle equal to the diameter 




Fig. 272. 



of the wheel to the points of the teeth. The slips are then cut to the cir- 
cumferences of the circular arcs described upon them. These two pieces 
so prepared are fastened together by screws, as in the ^g. 272 ; the piece 




Fig. 273. 



A, whose edge s f is an arc of the pitch circle, is fixed upon B, whose 
edge is an arc of the extreme circumference of the wheel, the space s s be- 
tween those edges being in breadth equal to the length of the teeth from 





Fig. 274. 



Fig. 275. 



tlio ])itc]i circle to tlic points. This done, a like templet is prepared for 
t]ni2rmi()7i (fig. 273). 
f Tlio i)air of tcmplots being tliiis prepared, two tracing points m ?)i arc 



DRAWING OF MACUINERY. 173 

inserted obliquely, and from behind into tlic piece D of the pinion teni- 
2)let. One of these points j)asse3 ont at the edge of the piece C, and the 
other at the edge of the piece D, and the templets are then placed uj)on 
each other, as shown in iigs. 27^ and 275 annexed, so that the circumfer- 
ence of the piece C, that is the pitch circumference of the pinion, shall 
meet tlie circumference of the piece A, that is the pitch circumference of 
the wheel. If in this position the templets be made to roll upon each 
other through a certain arc, pressing them at the same time slightly 
together, the tracing points will mark two epicycloidal curves upon the 
pieces A and B, as v and r; of these two curves, that marked v, which is 
traced on the face of the piece A, will be the curve of the lower portion or 
ilank, and that marked r will be the curve of the upper portion, or face of 
a tooth of the wheel. If now the thickness of the tooth be marked off on 
the edge of the piece C, that is, on the pitch circle, and the corresponding 
tracing j^oint be made to coincide with that point, the curves of the oppo- 
site side of the tooth will be formed by making the templets roll together 
in the contrary direction. A complete outline of a tooth of the wheel is 
thus described, to which a pattern tooth may be cut and used to shape the 
teeth by in making the wheel pattern. Instead of forming pattern teeth, 
many prefer to lay off the teeth by circular arcs coinciding approximately 
with the epicycloidal arcs found by the templets. 

The preceding mode of obtaining the curves of the teeth of a pair of 
wheels is faulty in as far as it gives a form of teeth smaller at the root than 
at the pitch circles, and also in the circumstance that a pair of wheels, 
formed in the manner described, will only work correctly with each other, 
and not with wheels of any other numbers, although of the same pitch. 
To obviate this, it is necessary to employ a modification of the ordinary 
practice. 

Let a thin slip of wood be provided, and let an arc of the pitch circle 
be struck upon it ; divide the slip into two portions through the line of 
this arc with a fine saw ; one part. A, will have a concave, and the other, 
B, a corresponding convex circular edge. Describe an arc del of the pitch 
circle upon a second board C D, upon which the pattern tooth is to be 
drawn. Fix the piece B upon the board, so that its circular edge may 
accurately coincide with the circumference of the arc el el. Ji. portion of 
a circular plate D is next provided, of the same radius which it is proposed 
to give to the generating circle : this plate has a fine tracing point atj? 
inserted into it, and projecting slightly from its under surface, and accu- 
rately coinciding with its circumference. Having set off the thickness of 
the tooth a c upon the pitch circle el el, so that twice this width increased 



174 



DRAWING OF MACHINEET. 



205^. 




rig. 276. 



by the clearance ^^licli it is desired the teeth should have, may be equal 
to the pitch, the generating circle D is made to roll upon the convex edge 

of B ; meantime the point at p will 
trace upon the board the curve of 
the faces of the tooth, having caused 
the point to coincide successively 
with the two jDoints a and c, and 
the circle to roll from right to left, 
and vice versa. 

Let the piece B be now re- 
moved, and the piece A applied 
and fixed, so that its concave 
edge may accurately coincide with 
the circular arc d d; then, with 
llie same circular plate D pressed 
against the concave edge of A, and 
made to roll upon it, the point at j?, which is made as before to coincide 
successively with the points a and c, will trace upon the surface of the 
board C D the two hypocycloidal arcs ah, cd, which form the flanks of 
the tooth. The complete tooth, thus formed, will work correctly with the 
teeth similarly described upon any other wheel, provided the pitch of the 
teeth be the same, and the same generating circle D be used to strike the 
curves upon the two wheels. In this manner the general forms of the 
teeth of the pair are determined ; and it only remains to cut them off at 
such lengths that they shall come into contact in the act of passing through 
the line of centres. 

It has already been observed, that having found the epicycloidal curves 
of the teeth by means of the templets, a common method of proceeding is 
to find, by trial, a centre and small radius, by which the arc of a circle 
can be described that w^ill coincide nearly with the templet-traced curve. 
A more commodious and certain method of determining the centre and 
radius of the approximate arc has been snpj^lied by Prof. "Willis in the 
construction of his Odontograjpli, manufactured by Messrs. Iloltrapfel vfe 
Co., London, for description of which see Apjpletons' Dictionary of Me- 
chanics, pp. 819, 820. 

Involute teeth. — Involute teeth have the disadvantage of being, when 
in contact, too much inclined to the radius, by wliicli an undue pressure is 
transferred to their axes. Tlieir mutual friction is thereby little affected, 
but lliat of tlie axes is increased, and their journals are more speedily 
woni. lUit tliey have at the same time the advantage of working with 



DEAWING OF MACHINERY. 



iO 



more accuracy under derangement and incorrectness of fitting, and any 
pairs of them will work truly together in sets within certain limits, how- 
ever different in diameters, the pitch being the same. 

To describe this cnrveT for the teeth of a pair, of which the radii of the 
pitch circles and pitch of the teeth are determined, we may employ the 
mode illnstrated by fig. 277. Let A and B be the centres of the pair, and 
eh he their pitch lines ; join A and B by a right line passing through c; 
from this last point draw c d, c d, perpendicular to the radials B d, A d, 
and cutting them in d and d; this line d d is then a common normal to 
the teeth in contact, and the perpendiculars A <f, B d, are the radii of the 
involute circles which form the acting; faces of the teeth. 



rN 



^ 



Fig. 277. 



Fig. 278. 



Having thus elucidated the principles of the operation of toothed geer- 
ing, and the form and proportions to be given to their parts, we proceed 
now to give complete drawings of different toothed wheels ; premising, 
that except in case of working drawings to full size, arcs of circles are 
almost invariably used by the draughtsman to describe the forms of the 
teeth ; and that in practice, too little attention is paid to the construction 
of epicycloidal and involute teeth. In many drawings details are unneces- 
sary, and spur geers are represented by pulleys and bevel geers as in fig. 
278, the pitch being written in in figures. 



PPcOJECTIOXS OF A SPUE ^VHEEL. 

PL XYII. — To draw side elevation (fig. 1), an edge view (fig. 2), and a 
vertical section (fig. 3), of a spur wheel with Stt teeth, and a pitch of two 
inches : 

Determine the radius of the pitch circle from table, page 161 ; against 
2 in. we find in the second column .6366 ; .6366 X 54= = 34.38, the diam- 
eter. Draw the central line A C B and the perpendicular D E ; on C as a 
centre, with a radius 17.19, describe the pitch circle, and divide it into 54 



176 DRAWING OF MACHINERY. 

equal parts. To eifect this division, without fraying by repeated trials 
that part of the paper on which the teeth are to be represented, describe 
from the same centre C^, with any convenient radius, a circle ah od; with 
the same radius divide its circumference into six equal parts, and sub- 
divide each sixth into nine equal parts, and draw radii to the centre Q' ; 
these radii will cut the pitch circle at the required number of points. 
Divide the pitch (2 in.) into 15 equal parts ; mark oif beyond the pitch 
circle a distance equal to 5^ of these parts, and within it a distance equal 
to 6J parts (see page 163), and from the centre C describe circles passing 
through these points ; these circles are projections of the cylinders bound- 
ing the points of the teeth and the roots of the spaces respectively. 

In forming the outlines of the teeth, the radii which, by their intersec- 
tions with the pitch circle, divide it into the required number of parts, 
may be taken as the centre lines of each tooth. The thickness of the tooth, 
measured on the pitch circle, is y\ of the pitch, and the width of the 
si)ace is equal to Z^-. These distances being set off, take in the com- 
passes the length of the pitch, and from the centre g describe a circular 
arc h i; and from the centre^', with the same radius, describe another arc 
7i 7c touching the former ; these arcs being terminated at the circles bound- 
ing the points of the teeth and the bottoms of the spaces respectively, form 
the curve of one side of a tooth. Tlie other side is formed in a similar 
manner, by drawing from the centre I the arc m n, and from the centre p 
the arc m o, and so on for all the rest of the teeth. 

The teeth having been thus completed, we proceed to the delineation 
of the rim, arms, and eye of the wheel. The thickness of the rim is usu- 
ally made equal to that of the teeth, namely, y'^j of the pitch, which dis- 
tance is accordingly set off on a radius within the circle of the bottoms of 
the spaces, and a circle is described from the centre C through the point 
q thus obtained. Within the rim, a strengthening feather q r, in depth 
about I of the thickness of the rim, is generally formed, as shown in 
the plate. The eye, or central aperture for the reception of the shaft, is 
then drawn to the specified diameter, as also the circle rejn-esenting the 
thickness of metal round the eye, which is usually made equal to the pitch 
of the wheel. 

To draw the arms, from the centre C, with the radius C u equal to the 
pitcli, describe a circle ; draw all the radii, as C L, wliich arc to form tlie 
centre lines of the arms, and set off the distance L v, equal to | pitch, on 
each side of these radii at the iniier circumference of the rim, and through 
all the points thus ol)tained draw tangents to the circle passing through 
u. The contiguous arms are rounded off into each other by arcs of circles, 



P L A T K X y I 



ITC 




r L A T E X A' 1 1 1 . 



176 




Cvj 



r\j\j'yvx, 





-V^JVlAj 



/^ 



~y 



DRAWING OF MACHINEEY. ITT 

whose centres are obtained by the following construction : — Taking, for 
example, the arc M P Q, it is obvious that its centre is situated in the 
straight line C E which divides equally the interval between two contigu- 
ous arms. Having fixed the point P (which should be at the same distance 
from t as the breadth of the feather at the back of the rim) draw through 
it a perpendicular H P to the line C E ; the question now becomes simply 
a geometrical problem, to draw a circle touching the three straight lines 
M ^, P P, and S Q. Divide the angle PPM into two equal parts by 
the straight line P O, which cuts C E in the point O, the centre of the 
circle required ; its radius is the line O M pei'pendicular to M ^. If, now, 
a circle be drawn from the centre C, with the radius C O, its intersection 
with the radii bisecting all the intervals between the arms will give the 
remaining centres, such as O', of the arcs required ; and the circle pass- 
ing similarly through M, marks all the points of contact M Q M', &c. To 
di-aw the small arcs terminating the extremities of the arms, set off upon 
the line C E, within the point r^ the required radius of the arcs, and from 
the centre C with a radius C %o describe a circle ; the distance r w being 
then transferred to the extremities of the arms at the points where they 
are cut by the circle, as at S a', will give the centres of the arcs required. 
Draw the central web of the arm by lines parallel to their radii, making 
the thickness about f inch for wheel of about this size. 

Having thus completed elevation, the construction of the edge view 
and vertical section becomes comparatively simple. Draw the perpendicu- 
lars F G and H I (figs. 2, 3) as central lines in the representations ; set off on 
each side of th'^se lines half the breadth of the teeth, and draw parallels ; 
project the teeth of fig. 1 upon fig. 2, by drawing through all the visible 
angular points straight lines parallel to A B, and terminated at either ex- 
tremity by the verticals representing the outlines of the breadth of the 
wheel ; project in lilve manner the circles of the hub ; lay off half length 
on each side of F G, and draw parallels to it. Tlie section (fig. 3) is sup- 
posed to be made on the line D E of the elevation ; project, as in fig. 2, 
those portions which will be visible in this section, and shade those parts 
which are in section. Tlie arms are made tapering in width, and somewhat 
less than the face of the wheel. 

Since the two projections (figs. 1 and 3) are not sufficient to exhibit fully 
the true form, a cross section of one of them is given at fig. 4 ; this section 
is supposed to be made by a plane passing through X X' and Y Y^ The 
points ?/, ^, in fig. 1, and corresponding lines in fig. 3, represent the edges 
of Icey-seat. 

12 



1Y8 DRAWING OF MACHINEKY. 



OBLIQUE PEOJECTION OF A SPUE WHEEL. 

Plate XIX. — ^In drawing a spur wheel or other object in an oblique 
position with respect to the vertical plane of projection, it is necessary, in 
the first place, to lay down the elevation and plan as if it were parallel to 
that plane, as represented at figs. 1 and 2. Then transfer the plan to fig. 
4, giving it the same inclination with the ground line which the wheel 
ought to have in relation to the vertical plane ; and assuming that the 
horizontal line A B represents the axis of the wheel, both in the parallel 
and oblique positions, the centre of its front face in the latter position will 
be determined by the intersection of a perpendicular raised from the point 
C^ (fig. 4) with that axis. Now it is obvious, that if we take any point, as 
a in fig. 1, the projection of that point on fig. 3 must be in the line a «, 
parallel to A B ; and further, this point being projected at a' (fig. 4), it 
must be in the perpendicular a' a; therefore the intersection of these two 
lines is the point required. Thus all the remaining points 5, (?, cZ, &c., may 
be obtained by the intersections of the perpendiculars raised from the 
points V ^ g\ d' ^ &c. (fig. 4), respectively, with the horizontals drawn 
through the corresponding points in fig. 1. It will also be observed, that 
since the points e andj?^, in the further face of the wheel, have their projec- 
tions in a and h (fig. 1), their oblique projections will be situated in the 
lines a a and 5 &, but they are also at 6 and/"/ consequently, the lines e a 
andy ^ are the oblique projections of the edges a' e' and V f . TTe have 
now to remark, that all the circles which, in the rectangular elevation 
(fig. 1), have been employed in the construction of this wheel, are pro- 
jected in the oblique viev\^ into ellipses, the length and position of whose 
axes may be determined without any difficulty ; lor since the plane F' G', 
in which these circles are situated, is vertical, the major axes of all the 
ellipses in question will obviously be perpendicular to the line A B, and 
equal to the diameters of the circles of which they are respectively the pro- 
jections ; and the minor axes, representing the horizontal diameters, will 
all coincide with the line A B. Thus, to obtain the ellipse into which the 
pitch circle is projected, it is only necessary to set off upon the vertical 
D E (fig. 3), above and below the point C, the radius of the pitch-circle, 
whoso horizontal diameter ij being at i' j' (fig. 4) is j^rojccted to ij (fig. 3) ; 
and thus having obtained the major and minor axes, the ellipse in question 
may easily be constructed. Tlie intersection of the horizontal lines gg^ h h, 
&c., with this circle gives the thickness of the teeth at the pitch line; and 
by projecting iu the same manner the circles bounding the extremities and 



PLATE XIX 



178 




PLAT E X X 



ITS 







-■-^ — =_^ _ 






























f 

i 

I t 

I 


cr 
























( 


0- 










t=i 























DRAWING OF MACIIINEKT. 179 

roots of the teeth, these points in each individual tooth maj be determined 
by a similar process. But since, in cases where strict accuracy is required, 
a greater number of points is necessary for the construction of the curva- 
ture of the teeth, two additional circles m n and o ]) may be drawn on fig. 
1, and projected to fig. 3, and the points of their intersection with the 
curves of the teeth projected to fig. 3, where the corresponding j)oints are 
indicated by the same letters. 

It is almost unnecessary to observe, that the instructions we have given 
for the drawing of the anterior face F^ G' of the wheel are equally appli- 
cable to the posterior H^ I', which is parallel to it, and in all respects the 
same ; the common centre of all the circles in it being at O' (fig. 4), is pro- 
jected to O in fig. 3. Hence, it will be easy to construct the ellijDses 
representing these circles in the oblique projection, and consequently to 
determine the points <?, /", X", (fee, in the curvature of the teeth ; observing, 
that as their centre lines converge to C in the front face, they all tend to 
O in the remoter surface, which is, however, for the most j^art concealed 
by the former. 

It would be superfluous to enter into any details regarding the con- 
struction of the oblique view of the rim, eye, and arms which are drawn 
upon precisely similar principles to those we have already so fully ex- 
plained. 

PEOJECTIOXS OF A BEVIL WHEEL. 

Plate XXI. — ^Fig. 1 is a face view, fig. 2 an edge view, and fig. 3 a 
vertical transverse section. We have explained (page 159) the determina- 
tion of the division of the angle of inclination of the axes of a pair of bevil 
wheels ; their size and proportion are to be determined by the rules given 
for spur-wheels ; thus, consider the base of the cone A B (figs. 2, 3) as the 
diameter of the pitch circle of a spur-wheel, and proportion the pitch, form, 
and breadth of teeth, according to the stress to which they are to be subjected. 

Having determined and laid down, according to the required condi- 
tions, the axis O S of the primitive cone, the diameter A B of its base, the 
angle A S O which the side of the cone makes witli the axis, and the 
straight lines A c", D o\ perpendicular to A S, and representing the sides 
of two cones, between which the breadth of the wheel (or length of the 
teeth) is comprised, the first operation is to divide the primitive circle, de- 
scribed with the radius A C, into a number of equal parts coiTesponding 
to the number of teeth oy lyitcK of the wheel. Then upon the section (fig. 
3), draw with the radius ^ A or ^ B, suppose-d to move parallel to itself. 



180 DRAWING- OF MACHINEET. 

outside the figure a small portion of a circle, upon which construct the 
outlines of a tooth M, and of the rim of the wheel, with the same propor- 
tions and after the same manner as we have explained in reference to 
spur-wheels ; set off from A and B the points a, cZ, and/*, denoting respec- 
tively the distances from the pitch line to the points and roots of the teeth, 
and to the inside of the rim, and join these points to the vertex S of the 
primitive cone, terminating the lines of junction at the lines D c', E o' ; 
the figure ah c d will represent the lateral form of a tooth, and the figure 
e df e a section of the rim of the wheel, by the aid of which the face view 
(fig. 1) may easily be constructed. 

The points a^ 5, <?, c?, and <?, having been projected upon the vertical 
diameter A' B^, describe from the centre C^ a series of circles passing 
through the points thus obtained, and draw any radius, as C L, passing 
through the centre of a tooth. On either side of the point L set off the 
distances L Ic, L Z, making up the thickness of the tooth M at the point, 
and indicate, in like manner, upon the circles passing through the points 
B' and d'^ its thickness at the pitch line and root ; then draw radii through 
the points i^ Z, ^, ^, m, &c., terminating them respectively at the circles 
forming the projections of the corresponding parts at the inner extremity 
of the teeth ; these radial lines will represent the rectilinear edges of all 
the teeth. Tlie curvilinear outlines may be delineated by arcs of circles, 
tangents to the radii g C and i C, and passing through the points obtained 
by the intersections of the radii and the various concentric circles. The 
radii of these circular arcs may in general, as in the case of spur wheels, 
be taken equal to the pitch, and their centres upon the interior and exte- 
rior pitch-circles ; thus the points g and % n and 6>, for example, are the 
centres for the arcs passing through the corresponding points in the next 
adjacent teeth, and mce versa. 

Tlie drawing of the teeth in the edge view (fig. 2), and of such portions 
of them as are visible in the section (fig. 3), is sufiiciently explained by in- 
spection of the lines of ]3rojection which we have partially introduced 
into the plate for this purpose. We have only to remark, that in the con- 
struction of these views, every point in the principal figure from wliich 
they arc derived is situated upon the projection of the circle drawn from 
the centre C, and passing through that point. Thus tlic points g and ^*, 
for example, situated upon the exterior pitch-circle, will be determined in 
fig. 2 by the intersection of their lines of j^rojection with the base A B of 
the primitive cone ; and the points h and I will be upon the straight line 
passing through a a (iig. 3), and so on. Farther, as the lateral edges of 
all the teeth in iig. 1 are radii of circles drawn from the centre C, so in 



PLATE XXI 



130 







i^LATE XXII. 



19fl 




DEAWING OF MACHINERY. 



181 



fig. 2 tliey are represented by lines drawn throiigli the various points 
found as above for the outer extremities of the teeth, and converging 
towards the common apex S ; while the centre-lines of the exterior and 
interior extremities themselves all tend to the points o and o' respectively. 
This circumstance will suggest a mode of materially simplifying the opera- 
tion of drawing the edge view of the teeth when the wheels are small, or 
executed to a small scale ; and in all cases it affords a means of testing 
the accuracy of the operations, if the method of projecting numerous points 
be adopted. 

Slceic-hevels. — When the axes of wheels are inclined to each other, and 
yet do not meet in direction, and it is j)roposed to connect them by a 
single pair of bevels, the teeth must be inclined to the base of the frusta to 




Fig. 279. 



allow them to come into contact. Set oE a e equal to the shortest distance 
between the axes, (called the eccentricity,) and divide it in c, so that a c 
is to e G as the mean radius of the frustum to the mean radius of that with 
which it is to work ; draw c m d perpendicular to a e. Tlie line cmd gives 



182 DE AWING OF MACHINERY. 

tlie direction of the teeth ; and if from the centre a^ with radins a c, a 
circle be described, the direction of any tooth of the wheel will be a tan- 
gent to it, as at c. Draw the line d e perpendicular to c m d^ and with a 
radius d e equal to c e describe a circle ; the direction of the teeth of the 
second wheel will be tangents to this last, as at d. 



SYSTEM COMPOSED OF A PINION DEIYING- A EACK. 

PL XXIII., fig. 1.— The pitch line M IST of the rack, and the primitiYC 
circle A B D of the pinion being laid down touching one another, divide 
the latter into twice the number of equal parts that it is to have of teeth, 
and set off the common distance of these parts upon the line M ^N", as many 
times as may be required ; this marks the thickness of the teeth and width 
of the spaces in the rack. Perpendiculars drawn through all these points 
to the solid part of the rack will represent the flanks of the teeth upon 
which those of the pinion are to be developed in succession. The curvature 
of these latter should be an involute A c of the circle A B D. Tlie teeth 
might be cut off at the point of contact d upon the line M I^, for at this 
position the tooth A begins its action upon that of the rack E ; but it is 
better to allow a little more length ; in other w^ords, to describe the circle 
bounding the points of the teeth with a radius somewhat greater than C d. 

"With regard to the form of the spaces in the rack, all that is required 
is to set off from M I^, as at the point <?, a distance slightly greater than 
the difference A <^ of the radius of the pitch circle, and that of the circle 
limiting the points of the teeth, and through this point to draw a straight 
line F G parallel to M IT. From this line the flanks of all the teeth of the 
rack spring, and their points are terminated by a portion of a cycloid A 5, 
Avhich, however, may in most instances be replaced by an arc of a circle. 
The depth of the spaces in the pinion obviously depends upon tlie height 
of tliis curved portion of the teeth ; their outline is formed by a circle 
drawn from the centre C, with a radius a little less than the distance from 
this point to the straight line, bounding the upper surface of the teeth of 
tlie rack. 



SYSTEM COMPOSED OF A KACK DRIVING A PINION. 

In this case the construction is in all respects identical with that of the 
preceding example, with this exception, that the form proper to be given 
to the teeth of the rack is a cycloid generated by a point A in the circum- 



V L A T K X X I I T 



Fi- 1. 




L!i:i£ai^' 



9iia®^ r 



PLATE X X I y . 



183 



Fi- 1 




DRAWING OF MACIIINEEY. 183 

ference of the circle A E C, in rolling on the line M X. The curvature 
of the teeth of the pinion is an involute as before. 

SYSTEM COMPOSED OF A ^^^IEEL AXD TANGENT, OE ENDLESS SCREW. 

Fig. 2. — In the construction of this varietj of geering, we must first fix 
upon the number of teeth in the wheel, and the distance of its centre from 
the axis of the screw. Then conceive a plane passing through the axis 
E F of the screw, parallel to the face of the wheel, and let C be the centre 
of its primitive circle. If now a perpendicular C G be drawn from C upon 
E F, and C A be taken as the radius of the pitch circle B A D of the 
wheel, the difi'erence A G will represent the radius of a cylinder, which 
maj be termed the primitive cylinder of the screw ; and a line M K drawn 
through A, parallel to E F, will be a generatrix of that cylinder, which 
will serve the purjDOse of determining the form of the teeth. 

The section having been made through the axis, the question obviously 
resolves itself into the case of a rack driving a pinion ; consequently the 
curve of the teeth, or rather thready of the screw should be simply a cycloid 
generated by a point in the circle A E C, described npon A C as a diam- 
eter, and rolling upon the straight line M I^. It is to be remarked, fur- 
ther, that the outlines of the teeth are helical surfaces described about the 
cylinder forming the screw, with the pitch A h equal to the distance, mea- 
sured upon the primitive scale, between the corresponding points of two 
contiguous teeth. Tliese curves have been drawn on our figure, but being 
for the most part concealed, they are expressed by dotted lines. The teeth 
of the wheel are not, as in ordinary kinds of geering, set perpendicularly 
to the plane of its face, but at an angle, and with sui-faces corresponding 
to the inclination and helical form of the thread of the screw. In some 
instances, the points of the teeth and bottoms of the spaces are formed of a 
concave outline adapted to the convexity of the screw, in order to present 
as much bearing surface as possible to its action. In this kind of geering, 
for obvious reasons, it is invariably the screw that imparts the motion. 

Fig. 3 represents an edge elevation of the wheel, projected as in pre- 
vious examples. 

SYSTEM COMPOSED OF AN ENTEENAL SPUR-WHEEL DRIVING A PIN-ION. 

PL XXY., fig. 1. — Tlie form of the teeth of the driving wheel is in this 
instance detei-mined by the epicycloid described by a point in the circle 
A E C, rolling on the concave circumference of the j)rimitive circle ]\I A X. 



184 DKAWrNG OF MACHINERY. 

The points of the teeth, are to be cut off by a circle drawn from the centre 
of the internal wheel, and passing through the point E, which is indi- 
cated, as before, by the contact of the curve with the flank of the driven 
tooth. 

The wheel being supposed to be invariably the driver, the curved por- 
tion of the teeth of the pinion may be very small. This curvature is a 
part of an epicycloid generated by a point in the circle MAE" rolling 
upon BAD. 



SYSTEM COMPOSED OF AN INTERNAL WHEEL DRIVEN BY A PINION. 

rig. 2. — ^This problem involves a circumstance which has not hitherto 
come under consideration, and which demands, consequently, a different 
mode of treatment from that employed in the preceding cases. The epicy- 
cloidal curve A a^ generated by a point in the circle having the diameter 
A O, the radius of the circle MAE", and which rolls upon the circle 
BAD, cannot be developed upon the flank A J, the line described by 
the same point in the same circle in rolling upon the concave circumfer- 
ence MAN; and for this obvious reason, that that curve is situated with- 
out the circle BAD, while the flank, on the contrary, is within it. It 
becomes necessary, therefore, in order that the pinion may drive the wheel 
uniformly according to the required conditions, to form the teeth so that 
they shall act always upon one single point in those of the wheel. This 
may be most advantageously effected by taking for the curvature of the 
teeth of the pinion the epicycloid A d described by the point A in the 
circle MAN, rolling over the circle BAD. It will be observed that, as 
in the preceding examples, the tooth E of the pinion begins its action upon 
the tooth E of the wheel at the point of contact of their respective primi- 
tive circles, and that it is unnecessary that it should be continued beyond 
the point c, because the succeeding tooth H will then have been brought 
into action upon G ; consequently the teeth of the wheel might be bounded 
by a circle passing through the point c. It is, however, one of the prac- 
tical advantages which this species of gccring has over wheels working ex- 
ternally, that the surfaces of contact of the wheel and pinion admit of 
being more easily increased ; and by making the teeth somewhat longer 
than simple necessity demands, the strain may be dift'iisod over two or 
more teeth at the same time. The flanks of the teeth of tlic wheel are 
ibrmcd l)y radii drawn to the centre O, and their points are rounded off 
to enable tliem to enter freely into the spaces of the pinion. 



P I. A T K X X \' 




P L ATE X X \' 



184 




135 



PLATE XXYir. 






DKAWIXCr OF MACHINERY. 185 



PROJECTIONS OF ECCENTRICS. 

Tlie term eccentric is applied in geneml to all such enrvos as arc com- 
posed of points situated at unequal distances from a central point or axis. 
The ellipse, the curve called the heart, and even the circle itself, when 
supposed to be fixed upon an axis which does not pass through its centre, 
are examples of eccentric curves. 

Tlie object of such curves, which are of frequent occurrence in machin- 
ery, is to convert a rotatory into an alternating rectilinear motion ; and 
their forms admit of an infinite variety, according to the nature of the 
motion desired to be imparted. Examples of their application occur in 
many arrangements of pumps, presses, valves of steam-engines, spinning 
and weaving machines, &c. 

Fig. 1, pi. XXV 11. — To draw the eccentrical symmetrical cin^ve called 
the heart, lohich is such as, lohen revolving loith a uniform motion on its 
axis, to communicate to a movcible jpoint K,a uniform rectilinear motion of 
ascent and descent. 

Let C be the axis or centre of rotation upon which the eccentric is 
fixed, an-d which is supposed to revolve uniformly ; and let A A' be the 
distance which the point A is required to traverse during a half revolution 
of the eccentric. From the centre C, with radii respectively equal to C A 
and C A', describe two circles ; divide the greatest into any number of 
equal parts (say 16), and draw through these points of division the radii 
C 1, C 2, C 3, ifcc. Then divide the line A A' into the same number of 
equal parts as are contained in the semicircle (that is, into 8 in the 
example now before us), and through all the points V , 2', 3', etc., di-aw 
circles concentric with the former ; the points of their intersection B, D, E, 
&c., w^tli the respective radii C 1, C 2, C 3, &:c., are points in the curve 
required, its vertex being at the point 8. 

It will now be obvious that when the axis, in its angular motion, shall 
have passed through one division, in other words, when the radius C 1 
coincides with C A^, the point A, being urged upwards by the curvature 
of the revolving body on which it rests, will have taken the position indi- 
cated by 1' ; and further, when the succeeding radius C 2 shall have 
assumed the same position, the point A will have been raised to 2', and so 
on till it arrives at A', after a half revolution of the eccentric. Tlie re- 
maining half A G F 8 of the eccentric, being exactly symmetrical with the 
other, will enable the point A to descend in precisely the same manner as 
it is elevated. It is thus manifest that this curve is fitted to impress a imi- 



186 DRAWING OF MACHINEEY. 

fonn motion upon the point A itself, but in practice a small friction 
roller is usually interposed between the surface of the eccentric and the 
piece which is to be actuated by it. Accordingly, the point A is to be 
taken as the centre of this roller, and the curve whose construction we 
have just explained is replaced by another similar to, and equidistant from 
it, which is drawn tangentially to arcs of circles described from the various 
points in the primary curve with the radius of the roller. This second 
curve is manifestly endowed with the same properties as the other ; for, 
supposing the point ^, for example, to coincide with A, if we cause the 
axis to revolve through a distance equal to one of the divisions the point 
y, which is the intersection of the curve with the circle whose radius is 
C 1^, will then obviously have assumed the position 1^ ; at the next por- 
tion of the revolution, the point g (which is such that the angle /* C ^ is 
equal to ^ C /") will have arrived at 2^, and so on. Thus it is plain that 
the point a will be elevated and depressed uniformly by means of the 
second curve, in the same manner as that denoted by A is actuated by 
the first. 

It is obvious that the movable point a must, in actual working, be held 
in contact with the surface of the eccentric ; this is generally accomplished 
by the action of a weight or of a spring ; but in forms similar to fig. 1, in 
which all the diameters, as A B, B F, D G, &c., are equal, two frictions 
connected and placed diametrically opposite each other may be used, 
which will be thus alternately and similarly impelled ; in many cases an 
eccentric groove is cut, and the friction roll or point a is made to slide in 
this groove. 

Fig. 2. — To draw a dovhle eccentrio ciirve^ which shall imjpart a uni- 
form motion of ascent and descent to the j^oint A, traversing an arc of a 
circle A A'. 

First, divide the given arc A A^ into any number of equal parts (8 in 
the present example), and from the common centre, or axis C of the eccen- 
tric, describe circles passing through each of the points of division 1', 2', 3', 
&c. Divide also the circle passing through O, the centre of the arc A A', 
into twice the immber of equal parts ; then taking up in the compasses the 
length A O, and placing one of the points at the division marked 1, do- 
scribe an arc of a circle, which will cut at B the circle drawn with the 
radius C V \ from the next point of division 2, mark off*, in the same man- 
ner, the point D in the circle whose radius is C 2', and so on. The points 
B, D, E, i^c, thus obtained, are points in the curve required, which, sup- 
posing the eccentric to revolve uniformly, will possess the property of 
coniinunicating to the point A a uniform motion of ascent and de- 



DRAWING OF MACHINEEY. 187 

scent along the arc A A^ This admits of easy demonstration. Tlie angle 
B C 1' is half of 2' C D, and consequently, when the point B has arrived 
at 1\ the radius C D, then coinciding with C B, will have passed through 
an angle equal to 1' C B, and again, at the next jDoint in the revolution, 
will coincide with C 2'. Tlierefore the portion B D of the curve Avill 
impel the given point through the arc 1' 2', in the same time and with the 
same velocity, as the part A B will have raised it from A to 1'. By a 
similar process of reasoning it will be manifest, that the angle 1' C B being 
just one-third of 3' C I, the point A will also traverse the space 2' 3' with 
a uniform motion. 

By a glance at the figure it will be seen that this curve is not symmetri- 
cal ; in other words, that the part A F E is not equal or similar to A D E. 
This may be accounted for by observing, that the arc h V, for instance, is 
equal to V B, and consequently the point h (which is determined by the 
intersection of the circle passing through V with the arc described from 
the centre 15) cannot be situated in the same position in relation to A as 
the point B, since the radius C A does not pass through V ; the same re- 
mark applies to all the other arcs, d 2', &c. It is not the less certain, 
however, that the part A F E of the eccentric will cause the given point 
to descend through the arc A^ A in the same uniform manner as it had 
been elevated by the part A D E. 

In the two preceding examples of eccentrics it has been shown, that 
the point A moves through equal spaces in equal times, both in ascending 
and descending. In some cases, however, this is by no means desirable ; 
thus, if the eccentric is destined to give motion to a mass of matter which 
offers considerable resistance, such a form would give rise to injurious and 
destructive shocks. In such cases, it is necessary so to regulate the cm-va- 
ture of the eccentric, that the point A shall move at the beginning and 
end of its stroke with diminished velocity ; and that for this purpose, 
the space A A should be unequally divided, as in the example which 
comes next nnder notice. 

-^ Eig. 3. — To draw a double and symmetrical eccentric curve, such as to 
cause the^oint A to move in a straight line, and with an unequal motion; 
the velocity of ascent 'being accelerated in a given ratio from the starting 
•point to the vertex of the curve, and the velocity of descent being retarded 
in the same ratio. 

Upon A A' as a diameter describe a semicircle, and divide it into any 
number of equal parts ; draw from each point of division V , 2', 3', &c., 
perpendiculars upon C A'; and through the points of intersection 1^, 2^, 3^, 
&c., draw circles having for their common centre the point C, which is to 



188 DRAWING OF MACHINEEY. 

be joined, as before, to all the points of division on the circle (A' 48.) 
The points of intersection of the concentric circles with the radii C 1, C 2, 
C 3, &c., are points in the curve required. 

Fig. 4. — To construct a double and symmetrical eccentric^ which shall 
produce a uniform, rectilinear motion, with joeriods of rest at the jgoints 
nearest to, and farthest from, the axis of rotation. 

The lines in the figure above referred to indicate sufficiently plainly, 
without the aid of further description, the construction of the curve in 
question, which is simply a modification of the eccentric represented at 
Fig. 1. In the present example, the eccentric is adapted to allow the 
movable point A to remain in a state of rest during the first quarter of a 
revolution B D ; then, dm-ing the second quarter, to cause it to traverse, 
with a uniform motion, a given straight line A A', by means of the curve 
D G ; again, during the next quarter E F G, to render it stationary at the 
elevation of the point A' ; and finally, to allow it to subside along the 
curve B E, with the same uniform motion as it was elevated, to its original 
position, after having performed the entire revolution. 

Fig. 5 represents an edge view of this eccentric, and fig. 6 a vertical 
section of it. 

Figs. T, 8, and 9, a Circular Eccentric. — ^These figures represent a 
model of a variety of the circular eccentric, which is the contrivance 
usually adopted in steam-engines for giving motion to the valves regu- 
lating the action of the steam upon the piston. The circular eccentric 
is simply a species of disc or pulley fixed upon the crank-shaft, or other 
rotating axis of an engine, in such a manner that the centre or axis 
of the shaft shall be at a given distance from the centre of the pulley. A 
ring or hoop, either formed entirely of, or lined with brass or gun metal, 
for the purpose of diminishing friction, is accurately fitted within project- 
ing ledges on the outer circumference of the eccentric, so that the latter 
may revolve freely within it ; this ring is connected by an inflexible rod 
with a system of levers, by which the valve is moved. It is evident, that 
as the shaft to which the eccentric is fixed revolves, an alternating recti- 
linear motion will be impressed upon the rod, its amount being determined 
by the eccentricity, or distance between tlic centre of the shaft and that 
of the exterior circle. Tlie throio of the eccentric is twice the eccentricity 
C E ; or it may be expressed as the diameter of the circle described by 
the point E. Tlic nature of the alternating motion generated by the cir- 
cular cccH^ntric is identical Avith that of the crank, which might in many 
cases be advantageously substituted for it. 

Fig. 8 is the edge view, fig. 1) the section of the eccentric, in this par- 



1S9 



rLz\TE XXYIJI. 




ISO 



PLATE XXIX. 




DRAWING OF MACHINEEY. 189 

ticular example, formed in a single piece, and wliicli can be applied only 
when tlie shaft to wliicli it is to be attached is straight and nninterrupted 
by cranks, &c. Tlie mode of representing the arm in fig. 9, which is a 
section on the line D P, is not strictly accnrate, bnt is a license frequently 
practised in similar cases, and which is attended with obvious advantage. 

In many machines, the eccentric is used for tlie raising of a weight a 
certain height and then letting it fall, as in the case of ore stampers, cloth 
beetles, trip hammers, and the valve rod of some steam engines. In these 
cases the eccentric may be considered as merely a single long tooth geer, 
in which commonly, on account of the uniformity of action, the wijying or 
rubbing surface is an involute curve, the boss of the eccentric being the 
generating circle. 

In practice, the term eccentric is generally confined to the circular 
eccentric ; all others, with exception of that last described, or wyjpers^ 
beino; called cams. 



DEAWIXG OF SCREWS. 

The screw is a cylindrical piece of wood or metal, in the surface of 
which one or more helical grooves are formed. The thread of the screw 
is the solid portion left between the grooves ; and the ^itch of the screw 
is the distance, measured on a line parallel to the axis of the cylinder, be- 
tween the two contiguous centres of the same thread. 

Projections of a triangular-threaded screiD and nut ^ pL XXYIII., fig. 1. 
— Havino: drawn the ccround line A B, and the centre lines C C^ of the 
figm*es, from O as a centre, with a radius equal to that of the exterior 
cylinders, describe the semicircle « 3 6 ; describe in like manner the semi- 
circle h c e with the radius of the interior cylinder. Xow draw the per- 
pendiculars a a" and 6 Q\ h ¥ and e c\ which will represent the ver- 
tical projections of the exterior and interior cylinders. Then divide the 
semicircle (^3 6 first described into any number of equal parts, say 6, and 
through each point draw radii, which Avill divide the interior semicircle 
similarly. On the line a' a!' set off" the length of the pitch as many times 
as may be required ; and through the points of di\dsion draw straight 
lines parallel to the ground line A B. Then divide each distance or pitch 
into twice the number of equal parts that the semicircles have been divided 
into, and following instructions already laid down (page 100), construct 
the helix a! Z' 6 both in the screw and nut. 

Having obtained the point V by the intersection of the horizontal line 
passing through the middle division of a' a with the perpendicular h V^ de- 



\ 



190 DRAWING OF MACHINERY. 

scribe the helix h' c' e\ which will represent the bottom of the groove. 
The apparent outlines of the screw and its nut will then be completed by 
drawing the lines V a\ a' h\ &c., to the curves of the helices ; these 
are not, strictly speaking, straight lines, but their deviation from the 
straight line is, in most instances, so small as to be imperceptible, and it 
is therefore unnecessary to complicate the di'awing by introducing the 
method of determining them with rigorous exactness. 

When a long series of threads have to be delineated, they should be 
drawn mechanically by means of a mould or tevi^let, constructed in the 
following manner : — ^Take a small slip of thin wood or pasteboard, and 
draw upon it the helix a' W 6 to the same scale as the drawing, and pare 
the slip carefully and accurately to this line. By applying this templet 
upon fig. 1, so that the points a' and 6 on the plate shall coincide with 
a' and 6 on the drawing, the curve of Z' 6 can be drawn mechanically, 
and so on for the remaining curves of the outer helix. Tlie same templet 
may be employed to draw the corresponding curves in the screw-nut by 
simply inverting it ; but for the interior helix a separate one must be cut, 
its outlines being laid off in the same manner. 

Projections of a square-threaded sci^ew and nut (fig. 2).' — ^The depth of 
the thread is equal to its thickness, and this latter to the depth of the 
groove. The construction is similar to the preceding, and will be readily 
understood from the drawing, the same letters and figures marking rela- 
tive parts. Tlie parts of the curve concealed from view are shown in 
dotted lines. 

It will be observed, that the heads and nuts of the screws are repre- 
sented as broken, which is done for economy of space. 

It is seldom necessary to delineate so exactly the outlines of screws, 
as they are generally drawn to a much smaller scale. Fig. 3 shows the 
simplest form by which a screw may be represented. Figs. 4, 5, 6, repre- 
sent a triangular- threaded, a square-threaded screw, and a serpentine, in 
which tlie helical curves are replaced by straight lines, and these forms 
will be found sufficiently exact and graphic for most of the cases occur- 
ring ill practice. 

Screws may have two, three, or even a greater number of threads, ac- 
cording to the velocity which their action may be required to produce. 
A dovltl ('-threaded screw is one in which the pitch of any individual helix 
iiic'lu(h's two thi-cads; a three-threaded scrav, ono. in whicli it oinbraccs 
three thj-cads, and so on. 

^Size and 'proportion of holts. — The diameter o^ tlie bolt depends, of 
course, on the strain to which it is to be subjected ; but since the tensile 



DRAWING OF MACHINERY. 



191 



strength of common bolts is reduced at least one quarter by tlie cutting of 
the thread, as a safe rule the tension ought not to exceed 4 tons on the 
square inch of section. It will be found economical often, where the bolt 
is long, to cut the thread on a larger wire, and weld the piece to a rod of 
the interior diameter of the screw. The section of the thread most ap- 
proved of for strength and easy motion of the nut is the equilateral tri- 
angle, thus A, the bevelled sides being equal between themselves and to 
the base. 



Diameter of 


Threads in 


Short diameter 


Diameter of 


Threads in 


Short diameter 


bolts. 


inch. 


of nut. 


bolts. 


inch. 


of nut. 


Inches. 




Inches. 


Inches. 




Inches. 


h 


12 


1 


If 


5 


3J 


1 


11 


ItV 


n 


H 


3| 


i 


10 


h% 


2 


H 


h% 


i 


8| 


h% 


2| 


4 


3| 


1 


8 


n 


2J 


4 


4 


n 


n 


2 


2| 


3^ 


^ 


n 


r 


n 


2i 


H 


ir\ 


ii 


6 


2A 


2f 


3 


i} 


1-^ 


5i 


21- J 


2J 


3 


m 


n 


5 


2f 


3 


3 


5? 



The thickness of the nut should be equal to the diameter of the bolt. 
The head of the bolt is usually square ; the nut may be of the same form, 
but as often is six-paned or six square. When the head of the bolt is in- 




Fig. 2S0. 




tended to ha flush or even with the surface of the j^iece into which the bolt 
is inserted, the inside of the head is made conical like the common wood- 



192 DRAWING OF MACHmERY. 

screw, or pyramidal, and it is then said to be countersunk. When bolts 
are employed in wood, washers are usually placed beneath the nut and 
head, to give a more extended bearing surface. 

HOOKS. 

Figs. 280 and 281 represent two wrought iron hooks, in which the 
material is distributed according to the strain to which the parts may be 
subjected. The following are the proportions on which fig. 280 is con- 
structed : — ^Assuming the neck of the hook as the modulus or 1, the diam- 
eter of journals of the traverse are 1.1 ; width of traverse at centre, 2 ; 
distance from the centre of the hook to the centre of the traverse, Y.5 ; 
interior circle of the hook, 3.4 ; greatest thickness of the hook, 2.8. As- 
suming (fig. 281) the diameter of the wire of the chain as 1 : interior circle 
of hook is 3.2, and greatest thickness of hook, 3.5. 

FRAMES. 

Plate XXX. represents the application of iron in the frames of tools. 

Fig. 1 represents the cam-punch and shear ; in this case, the force 
exerted whilst the machine is in the operation of punching or shearing, 
tends to open the jaws a a; and the tendency increases with the depth of 
the jaw, the strain obviously being the greatest at the inmost part of the 
jaw. The frame consists of a plate of cast iron, with two webs around its 
edges; the front web being subjected to a tensile strain, should be in the 
area of its section about six times that of the rear web which is subjected 
to a compressive force. 

Fig. 2 is the side frame of a planing machine. The force here exerted 
is horizontal against the cutter, which can be raised or lowered at pleasure, 
according to the magnitude of the work to be planed ; the upright has, 
therefore, to be braced, which is done in a curved form for beauty of out- 
line. 

Fig. 3 is a common jack-screw, in which the pressure is vertical ; the 
base is made extended to give it stability. 

Fig. 4 is a plan of the top plate, and fig. 5 the elevation of a hydraulic 
press. The top and bottom plates and platen are cast iron, the four i-ods 
arc wrought iron ; tlie strain upon the rods is tensile, and it is only neces- 
sary to give them such a size as to resist securely the power which nuiy be 
recpiired on tlie press. The plates arc beams, supported at the four cor- 
ners ; subjected to a breaking strain, it will be evident that the bottom 



PLATE XXX 



Fi- 4. 



li;r 1. 




Fiir. 5. 



.^^^MMMMMMMMMS^z^ 




-~Y 



U^ 



cm 



1 



-I 



<r^^ 




^= 



G 






DRAWING OF MACHINEEV. 193 

plate is the strongest, as in this case the bottom of the plate being sub- 
jected to a tensile strain, is a flange or platen, and affords more material 
remote from the neutral axis than the ribs of the upper plate which are 
subjected to the tensile strain. Tiie movable platen is braced by triangu- 
lar wings or flanges radial from the piston. In this particular case tlie 
cylinder is cast iron hooped with wrought iron ; it is very common to 
make the whole cylinder wrought iron. 

Principle of the action of the hydrostatic press. — Let A B C D (fig. 282) 
epresent a vertical section of a cylindrical vessel filled with 
an incompressible non-elastic fluid, as water for instance ; let 
E and F be two pistons of different magnitudes connected with 
the cylinder, and fitting closely their respective orifices ; now, 
whatever pressure be exerted by the piston F on the fluid ^''°- 2^^- 
in the cylinder, it will be repeated on the piston E as many times as the 
area of the small piston is contained in the large piston ; that is, if the 
area of F was 1 square inch, and the pressure exerted 10 lbs., and the pis- 
ton E 100 square inches, then the pressure on E would be 10 x 100, or 
1000 lbs. F corresponds to the plunger of the force-pump, E to the piston 
or ram of the press. The thickness of metal of the cylinder, if of cast-iron, 
should not be less than one-half the diameter of the ram. Adopting this 
as the rule, to find the entire pressure in tons which a cylinder can sustain, 
the diameter of the ram being given : 

Multiply the square of the diameter in inches by 3, and the product 
will be the pressure in tons. 

Or, the pressure in tons being given : 

Divide the given pressure in tons by 3, and the square root of the quo- 
tient will be the diameter of the piston in inches. Thus, the diameter of 
the piston being 10 inches, the thickness of metal 5 inches, the pressure 
might be 10 X 10 X 3 =r 300 tons. 

Figs. 6 & 7 represent a housing for rolls. Tlie screw a presses down 
npon the top of the box of the journal, and the effect is a tensile strain on 
the sides of the frame ; but it must be remarked, that frames of this sort 
are subject to percussive and intermittent strain vastly exceeding the mere 
tensile strain, and proper allowance is to be made for this ; and it is much 
better to depend in part on mass or on dead weight of material to resist 
such strains than upon cohesive strength merely. 

PL XXXI. represents the elevation of the frames of three classes of 
American marine engines. 

Figs. 1 and 2 represent the frame-work of the Neio World. It is com- 
posed of four pieces of heavv pine timber d d^ Avhich are formed into tAvo 
13 



194 DRAWING OF MACHINERY. 

triangles, and inclined sliglitly laterally to each Other (fig. 2) ; their lower 
ends rest on the keelsons, and npon their npper extremities are ^Dlaced the 
pillow-block c of the working beam. They are solidly fastened together 
and to the boat by numerous horizontal and diagonal timbers, which are 
secured by wooden knees and keys, and are heavily bolted. Tlie two front 
legs are bolted to flanges cast on the sides of the condenser, and the other 
end of the framing is attached to a large mass of timbers, which support 
the shaft pillow-block h. The framing is further steadied by two addi- 
tional timbers, and rods running from the beam pillow-blocks outside the 
shaft to the keelsons of the boat ; a represents the guides, which are bolted 
at the bottom to the cylinder flange, and retained in their vertical position 
by wrought iron braces connected with the framing. The entire fastening 
of the engine and its framing is so disposed as to reduce all the strains to 
direct ones of extension or compression on the fibres of the iron and wood 
employed in the construction. The height of the frame is 46 feet, width 
at bottom 31 feet. 

Fig. 3 represents the side elevation of the frame of the side lever ocean 
steamer Pacific. In this frame the two large hollow pillow-blocks which 
sustain the shaft on each side of the cranks are supported by four wrought 
iron columns G G on the forward extremity of the bed-plate I, the centre 
of the shaft being 23 feet above the keelson. The pillow-blocks thus sup- 
ported are connected by two strong inclined braces D to the cylinder, by 
means of solid facings cast with it on each side of the steam opening. The 
columns are connected by horizontal braces A A, composed of hollow tubes, 
through which bolts pass, and the frames of the two engines are connected 
at the same points by similar tubes, and also two diagonal horizontal braces 
cast together. Similar braces C C are used to connect each extremity of 
the pillow-blocks, and the two engine frames are connected by a horizontal 
wrought iron cross. To resist the tendency of the engines, in the rolling 
of the ship, to press the outer bearings, there arc in a vertical transverse 
plan three wrought iron cross or diagonal braces F between the pillow- 
l)locks and bed-plates. Four cross braces II and J connect the extremities 
of tlic cylinder and the frame. Tlie cylinders are also connected by a 
liorizontal tubular brace. It will be tlms seen that this frame is a sys- 
tem of l)racing and cross-bracing, in which the material is most economi- 
cally disposed to resist the various strains. 

The bed-plate consists of a single casting, 32 feet long and feet broad, 
which is securely fostencd to the keelsons and ship's bottom ; the diameter 
of the cylinder is 0(; inches, and the stroke feet. 

Fig. 4 represents tho si(U^, view of the frame (^f the inclined engines of 



PLATE XXXL 



194 




195 



PLATE XXXII. 



^. 




DRAWING OF ]MACIIINERY. 195 

tlie war steamer Susquehanna. The cylinder A is rested between two tri- 
angular frames, on the inclination of the longest side, and is securely bolted 
to each frame. The two frames are connected together with braces similar 
to those of the Pacific^ and the whole is securely bolted on to the keelsons 
and bottom of the ship. In this illustration, the main pieces of the frame 
arc made of boiler iron, constructed like box girders ; but in smaller en- 
gines, it is usual to make these parts of wood. The diameters of the cylin- 
ders are 5 feet 10 inches, the length of stroke 10 feet. 

PL XXXII. — Fig. 1 represents the working-beam of the New World. 
It is composed of a skeleton frame of cast iron, round which a wrought 
iron strap A is fixed. Tliis strap is forged in one piece, and its extreme 
ends are formed into large eyes, which are bored out to receive the end 
journals. The skeleton frame is a single casting, and contains the eyes for 
the main centre and air-pump journal ; the centre hub is strengthened by 
wrought iron hoops a a^ which are shrunk upon it. At the points of con- 
tact of the strap and skeleton, key-beds are prepared, into which the keys 
are carefully fitted and tightly driven ; the keys are afterwards riveted 
over at both ends, which retains them in their places, as well as the strap 
on the skeleton frame. The strap is also secured to the frame by straj3s 
h h and keys. The skeleton frame is still further braced by wrought iron 
straps C C, which tie the middle of the long arms of the cross to the ex- 
tremities of the shorter ones. This form of beam is that usually adopted 
for the engines of eastern American river boats ; the proportions are some- 
what varied, but the form is identical. The following are the dimensions 
of ^ur illustration : — From centre to centre of end journals, 26 feet ; this 
is somewhat less than the usual proportion to length of stroke, being but 
slightly less than double the stroke ; length of centre hub, 26 inches ; 
diameter of main centre eye, 15 f ; of eye for air-pump journal, 6f ; of end 
journal, 8{ inches. 

Fig. 3 represents a side elevation ; fig. 4, a plan ; and fig. 5 a section of 
a cast iron working beam of an English stationary engine. It will be per- 
ceived that the outline of the beam is a parabola, it being in efiect a beam 
supported at the centre and loaded at the extremities. 

From the following table of practical examples from " Architecture of 
Machinery," w^e would assume as a safe rule for land engines, that the 
depth ut centre should be the diameter of the cylinder, and the length of 
beam three times the length of stroke. Hence we can construct the out- 
line, having for the vertex the extremity of the beam and the j^oint B in 
the curve at the centre. The sectional area may be estimated from rules 
already given, knowing the load at the extremity, that is ; the pressure on 



196 



DRAWING OF MACHINEET. 



tlie piston, the weight of the same and its connections, and also the force 
required to drive the air-pump, estimated at the extremity of the lever. 
As an engine is subject to shocks, the load should be estimated as six 
times the absolute load. " Five per cent, of the nominal power of the 
engine may be considered the maximum of power required to drive the 
air-pump." — Ed. Tredgold. 



Diameter of 

cylinder. 


Length of stroke. 


Description of 
work. 


Length of beam 
from centre. 


Depth at centre. 


Sectional area. 


inches. 


ft. in. 




ft. in. 


inches. 


square inches. 


m 


8 


Eolling, 


12 4 


48 


240 


401 


7 


Pumping, 


10 4 


36 


162 


m 


6 9 


Blowing, 


9 6 


3Si 


96i 


861 


6 3 


Boiling, 


9 8 


30 


60 


24f 


5 


Mill work, 


8 


25 


50 


18i- 


4 


u 


6 10 


22i 


50 


42 


4 


Marine, 


6 3 


23 


138 ■ 


42 


4 2 


" 


6 6 


27 


216 


32 


3 


" 


5 


22 


132 



Figs. 6 and 7 represent a side and a front elevation of a crank, such 
as is usually adopted on the engines of American river boats. The main 

body of the crank is of cast iron, with two horns 
a a projecting from the central hub, and the 
whole is bound with a strap of wrought iron. 
It is evident that this form of crank gives the 
greatest amount of strength with the least ma- 
terial, and belongs to the same class of construc- 
tion as the working beam (fig. 1). The eye of 
the crank is usually made one-fourth the diam- 
eter of the cylinder. The table from Eedten- 
bacher here inserted gives the relative sizes of 
central and end eyes of cranks, depending on 
the proportion between the length of crank and 
the diameter of central eye. The first column 
exhibits the number of times the diameter of eye is contained in the length 
of crank ; the other columns exhibit the diameter of crank-pin. 

From this table may be determined for any crank the diameter of 
citlier eye, one being known, and the length of the crank. 

Figs. S, J), a side view and front elevation of a wrought iron crank and 
their j)ractical proportions ; the eye for the crank-pin is a slightly conical 
Jiole, and llie ])iii is iiukU^ of a corresponding taper. 

PI. XXXIII. represents steam-engine connecting-rods and their details. 
Figs. 1, 13, represent the front and side elevation of a cast iron connect- 



DIAMETER OF EYE, BEING UNITS. 




For wrought 


Cast iron 




iron shaft. 


shafts. 


2 


0.85 


0.62 


3 


0.69 


0.51 


4 


0.60 


0.44 


5 


0.54 


0.39 


6 


0.49 


0.36 


7 


0.45 


0.33 


8 


0.42 


0.31 


9 


0.40 


0.29 


10 


0.38 


0.28 


11 


0.36 


0.26 


12 


0.34 


0.25 


13 


0.33 


0.24 



197 



PLATE XXXIII. 




DRAWING OF MACHINERY. 197 

ing-rod. It is a bar, strengtliened throughout the greater part of its length 
by four ribs or feathers, whose outlines in the direction of their length are 
parabolic curves. Its upper extremity is formed into two projecting arms, 
upon each of which a close wrought iron strap A is fixed by means of a 
key or cotter c. These straps are provided and formed for the reception 
of the brass bushes a and J, which are accm'ately fitted to the journals or 
bearings of the cross head. 

The lower end of the connecting-rod is made of a form suitable for the 
reception of the brasses, and other adjusting mechanism necessary for the 
purpose of acting freely, but without play, upon the pin of the crank. 
In the present example, the end of the crank-pin is concealed by a slight 
brass cover or disc, fixed to the connecting-rod by two small, screw pins, 
which serves to j)rotect the working surfaces from dust, and imparts an 
elegant finish to the whole. 

In fig. 3 the ends of the connecting-rod are represented upon a scale 
of double the magnitude of the preceding figures. One of the upper links 
or straps, with its adjusting apparatus, is supposed to be cut by a vertical 
plane passing through the axes, so as to expose the interior arrangement. 
This section exhibits distinctly the mode of fixing the links upon the arms 
of the connecting-rod by means of the cotters c, c^ and projecting discs <?, e^ 
cast upon the arais ; as also the contrivance for retaining the brasses a and 
h in their places. Fig. 5, which is a vertical section, shows the corre- 
sponding provisions for the lower end of the rod ; a small oblique hole for 
the introduction of oil will be observed in the upper brass m, while the 
lower n is formed with a spherical projection entering a concave recess in 
the cast iron, for the pui-pose of preventing its displacement by the friction 
of the crank-pin, which is regulated and adjusted by the cotter d. 

Fig. 6 is a horizontal section on the line a 5, showing the form of the 
body and feathers of the rod. / 

Fig. 4 represents the end of a connecting-rod, in which the arrange- 
ment for tightening the brasses consists of a gib h and cotter a. The small 
end of the cotter is made with a screw, wdiich passing through a lug on 
the gib, is fitted with a nut, by means of which the cotter is adjusted and 
retained in any position required. 

Figs. 7 and 8 represent the side and front elevation of a wrought iron 
connecting-rod, such as are generally used on American river boat engines. 
The extremities are fitted with brasses, straps, gibs, and cotters, similar 
to those already described. Tlie peculiarity over the general English con- 
struction is the economy of material, and the means adopted to give the 
required stifihess. It consists of a double truss brace a a of round iron, 



198 DRAWING OF MACHmEEY. 

wliich. is fastened hj bolts to the rod near each end ; struts h h, cut with a 
screw, and furnished with nuts pass through the centre of the brace, bj 
which means the braces are tightened. 

The length of connecting-rods, as recommended bj English mechanics, 
is three times that of the stroke ; in this country shorter connecting-rods 
are used, twice the length of stroke being not an unusual proportion. The 
connecting-rod at its smallest part near the extremities is of the same diam- 
eter as the piston-rod ; the boss in the centre is from 1 to 2 inches more. 

ON THE LOCATION OF MACHINES. 

In the arrangement of a manufactory or workshop, it is of the utmost 
importance to know how to place the machinery, both as to economy of 
space and also of working. Where a new building is to be constructed for 
a specific purpose of manufacture, it will be found the best to arrange the 
necessary machines as they should be, and then build the edifice to suit 
them. For defining the position of a machine, we merely need in outline 
the space it occupies in plan and elevation, and the j)osition of the driven 
pulley or geer, and of the operative. To illustrate this subject, we have 
selected a two- story weaving room, of which ^g. 283 is an elevation and 
plates XXXIY, and XXXY. plans. 

In this example the building is rectangular, of a width and length to 
accommodate the required machinery. The illustration is confined to a 
few rooms in one angle, the rest being but a repeat of the same. The tim- 
bering and planking are the same as adopted at all our large manufacturing 
places. Beams 14 to 16 inches deep, and of little less width, placed from 
8 to 9 feet apart from centre to centre, and floored with 3 to 4 inch 
plank dowelled or matched, with top floors and bottom sheathing. Tlie 
form of construction being fixed, and the size of the building being deter- 
mined for the number of looms, knowing the space they require for the 
machines and the alley ways ; lay down the outlines of the building, and 
dot in, or draw in red or blue, the position and width of beams. Tliis last 
is of importance, as it will be observed (fig. 283), tliat no driving-pulley 
can come beneath the beam, and also that this is the i30sition for the 
hanger. Lay off now the widtli of the alleys and of the machines. The 
first alley, or nearest the wall, is a back alley ; that is, where the operative 
docs not stand, and so on alternate alleys. Draw the lines of shafting cen- 
tral to the alleys, as in this position the belts arc least in the way. One 
operative usually tends lour looms ; they are therefore generally arranged 
in sets of four, two on each side of the alley, being placed as close to each 



DRAWING OF MACHINERY, 



199 



Other as possible, say one incli between the lathes, a small cross alley behig 
left between them and the next set. Lay off now the required alley at the 
end of the room, and space off the length of two rows of looms with alleys 
at the end of alternate looms, and mark the position of the pulleys. It 




will be observed that looms are generally rights and lefts, so that the 
pulleys of both looms come in the space where there is no alley. Should 
the pulley come beneath a beam, the loom must be either moved to avoid 



200 DKAWINa OF MACHINERY. 

it, or tlie pulley may be shifted to the opposite end of the loom. Parallel 
with the pulleys on the looms draw the driving-pulleys on the shafts, that 
is, h parallel with \ h with &, f with /*, and so on. Proceed now to draw 
the third and fourth row of looms, since the second and third rows are 
driven from the same shaft ; if they are placed on the same line, it will be 
impossible to drive both from the same end, and as this is important, we 
move the third row the width of the pulley 5, and for the sake of unifor- 
mity, the fourth row also. Lay off now the length of looms and position 
of pulleys as before, and parallel with the pulleys the driving-pulleys on. 
the shaft, that is c against <?, f against /*, and so on. Having in this way 
plotted in all the looms, every alternate set being on a line with the third 
and fourth row, we proceed now to lay down the position of the looms in 
the floor above ; and since for economy of shafting it is usual to drive 
from the lines in the lower rooms, to avoid errors, interference of belts and 
pulleys, it is usual to plot the upper room on the same paper or board 
as the lower room, using either two different colored inks, or drawing the 
machines in one room in deep and in the other in light line, as shown in 
plate XXXy. If the width of the rooms are the same, the lateral lines 
of looms and alleys are the same, and it is only necessary, therefore, 
to fix the end lines. ISTow, as the first loom in the outer row of looms, 
in the lower room, occupies for its belt \h.Q position h on the shaft, 
the loom in the upper room must be moved either one way or the 
other to avoid this ; thus the position i of the pulley on the loom must 
be made parallel to the pulley i on the shaft, so in the other looms a to a^ 
etoe^d to d^ and h to h. 

Besides the plan, it is often necessary, and always conveniet, to draw a 
sectional elevation (as in fig. 283), of the rooms, with the relative positions 
of the driving pulleys and those on the machines, to determino suitably 
the length of the belts, and also to see that their position is in every way 
the most convenient possible. For instance, in the figure, one of the lower 
belts should have been a cross belt, and one of the upper ones straight : 
now had the belts to the second row of looms in the upper story, been 
drawn as they should have been, straight, the belt would have interfered 
a little with the alley, and it would have been better to have moved the 
driving shaft a trifle towards the wall. 

From this illustration of the location of machines, knowing all the re- 
quirements, in a similar way any machinery may be arranged with economy 
of spaces, materials, power, and attendance. Tlicse two last items arc of 
the more importance as they involve a daily expense, where the others are 
almost entirely the first outlay. 



*'^*-- 



PLATE XXXIY. 




IM.ATK XXXV. 



^^"^^^^^^^^^x^^;^^:^ 



id 



f$^* 



11 



Lii_J 



nn ! 



r^~^ 



^ 









rvv^lL 



.iT 



i(> 









-^IJI 



_rzrL 



r ~i 



i_^ 



ron" 



I L_._i 



r~n 



DRAWING OF MACHINERY. 



201 



MACHINES. 



Fig. 234. 



'"-/ 



Ilitlierto we have confined ourselves to geometrical projections, and to 
the delineation of j^arts of machines, we now proceed to give representa- 
tions of complete machines, taken from actual constructions, together with 
a full description of their working, so that they may be not only copies for 
tlie mere draughtsman, but also examples to aid the engineer. Working 
Drawings are drawings in detail of 
the parts of a machine, generally in 
mere outline, or with but sufficient 
of shade to distinguish circular from 
flat parts. The drawings already 
given of parts of machines may be 
taken as illustrations of working 
drawings, but on a much smaller 
scale than in actual practice, it be- 
ing usual to make them as near 
full size as possible with the di- 
mensions written in, as many views 
of the parts in plan, elevation and 
section are given as may fully ex- 
plain the construction. 

The working of a machine may often be illustrated by a very few lines 

— or a skeleton drawing, of 
which fig. 284 may be taken 
as an example, which rep- 
resents the side elevation of 
Maudsley and Field's direct 
action double cylinder ma- 
rine engine, A and A^ being 
the cylinders, L the T plate 
connecting the two pistons, 
E the crank, F the wheel 
shaft, and II the air pump. 
Fig. 285 is the same engine 
with the cylinders, T plate 
and frame in outline, and 
the beams and connecting- 
rods in skeleton. 





^iT 












If 








- 










J 


A 


a2 


II 










1 



Fig. 2S5. 




202 DEAWING OF MACHINEEY. 

Plate XXXYI. is a sectional elevation through the centre of the cyl- 
inder of a Cornish Engine. XXXYII. is a front elevation showing the 
valve geer. 

A is the cylinder, enclosed in a jacket or casing of cast-iron, of such a 
diameter as to leave a clear space of one inch all around between it and 
the outside of the cylinder. This space communicates with the boilers by 
a pipe a / the boilers being so placed that the water line is at a lower 
level than the bottom of the cylinder, the water produced by condensa- 
tion in the jacket finds its way back again to the boilers. The space d^ 
under the bottom of the cylinder is kept constantly filled with steam by 
a branch from the pipe a. 

In order to prevent radiation from the outside of the steam case, the 
whole is surrounded with an outer covering of wood (shown by dotted 
lines 5,) leaving an interstice between the wood and the iron. This space 
is filled with some bad conductor of heat. The cylinder cover is fitted 
with a false lid or cap c, enclosing a thick layer of sawdust or other bad 
conductor, over the whole of the metal of the cover. 

The stuffing box on the cylinder cover, through which the piston rod 
passes, is of ]3eculiar construction. In the middle of the packing, and divid- 
ing it into two horizontal layers, a small chamber is formed by two brass 
rings G c^ kept apart by small distance pieces. Into this chamber steam is 
admitted by a small pipe from the jacket of the cylinder, the object of the 
arrangement being to prevent any leakage of air into the cylinder. 

C is the plug rod for working the valves and cataract. The plug rod 
is sometimes single and sometimes double, the latter in this instance. The 
two rods mxarked C 2, and C 3, 4, fig. 2, are connected at their upper ex- 
tremity by joints with the back links of the parallel motion, and work 
through guides i i. The rod C 3, 4, is lengthened at its lower extremity 
for the purpose of working the lever of the cataract. 

D is the top nozzle, shown on a larger scale and more fully figs. 3, 4, 
PLXXXYIIL It contains three valves, viz. First; Y% the goveriior ov 
regulating valve, for regulating the admission of steam into the chamber 
Teh oi the nozzle. The motion of the governor valve is commanded by a 
handle placed within reach of the engine-man, and connected by a rod/*' 
and lever </', with the stalk of the valve. The rod has a micrometer screw 
on its upper end, which works into a united socket attached to the end of 
tlie lever g\ and thereby raises or lowers the valve. Second; Y^, the 
steam valve, for admitting steam into the cylinder. When this valve is 
raised (the governor valve being supposed open also), the steam finds a 
passage through it, from the nozzle chamber 7l Z', into the sj^ace I, and 



PLATE XX.XYT. 




PLATE XXXYII. 



202 



4 



m 



.3^ 



Mm 






jr 



An 



202 



PLATE XXXYIII. 



e" 



^ 




^7-^ 



^^h 



1 

r 

3< 




P 




M^ 



m 



i 



f11 



'%r 



15 



;;;;s 






»^ 



^Jl 



^r4^ 





DRAWING OF MACHINERY. 203 

thence by tlie upper steam port m into the upper part of the cylinder ; the 
piston therefore descends, and the engine commences its in-door stroke. 

Third ; Y ^5 situated in the middle of the nozzle, is the equilibrium valve. 
WJien this valve is opened the steam above the piston finds its way by the 
equilibrium pipe E, and the lower port ?i, into the lower part of the cylin- 
der, until an equilibrium is restored between the pressures above and be- 
neatli the piston, which is then drawn upwards by the preponderating 
weight of the rods hung at the outer end of the beam. The top nozzle is, 
like the cylmder jacket, enveloped in an external casing. 

F is the bottom nozzle ; it contains Y% the exhaustion valve, for open- 
ing or closing the communication between the lower part of the cylinder 
and the condenser. The nozzle chamber above the valve communicates 
with the cylinder by the lower port n, while to the bottom of the nozzle, 
under the valve, is attached the eduction pipe H. The four valves are 
double-beat valves, so called from their having two beating faces. 

G is the cataract, fully shown in two sections, figs. 1, 2, PI. XXXYIII. 
the use of which is to regulate the period of opening the steam and ex- 
haustion valves. It consists of a barrel in which works a plunger, being 
in fact simply a small plunger forcing pump. The iiilet is by a valve g' 
opening freely upwards, but the outlet is contracted at pleasure by a move- 
able plug cT. Tlie pump is placed in a cistern of water G, and the plunger 
IS attached by a joint to the arm e' of the lever e' f. When the plug rod 
C^.^, has descended nearly to the bottom of its stroke, a tappet (/' upon 
the lower part of it strikes the end of the lever and thus raises the plunger, 
the water entering freely through the valve c'. When the stroke is finished, 
and the plug rod begins to ascend, the tappet g' quits the lever, and the 
weight h\ which is fixed upon the arm e\ and has been raised by the pre- 
ceding motion, becomes in its turn the motive power, tending to expel the 
water from the pump, by forcing the plunger down. But the inlet valve 
before c\ is closed, and the only exit for the water is by the aperture left 
round the regulating plug d'. It is evident therefore that the interval be- 
tween the time the tappet g' leaves the cataract lever, and the commence- 
ment of the next stroke, depends upon the time occupied by the descent 
of the cataract plunger, and ultimately upon the degree of opening given 
to the regulatmg plug d. This can be adjusted mtli great nicety by means 
of a micrometer screw and handle V y (PI. XXXYII.,) connectjed with 
the regulating plug by the rod t and lever I. 

Tlie parts of the cataract being all circular, the two sections given at 
right angles to each other, are all that are necessary to complete its con- 
struction, and in this respect, therefore, are sufficient working drawings. 



204 DRAWING OF MACHINERY. 

Plate XXXIX. fig. 1, is a longitudinal section of a locomotive boiler, 
and fig. 2 an interior view of the smoke box. 

The steam space C at the fire end of the boiler is half globe shaped, 
and surmounted by a dome. The object of the dome is to cany the steam 
as high as possible above the water line before its introduction into the 
steam pipe_^, in order that the water held in suspension near the surface of 
the water, may not be carried over into the cylinders. The steam pipe trav- 
erses the length of the boiler, and in the smoke box branches off to each 
cylinder ; I is the regulating or throttle valve, worked by the handle which 
passes out through a stuffing box in the end of the boiler ;^' is the fire box.' 
surrounded on all sides except at bottom with a water space ; the top or 
crown sheet of the fire box is strengthened by pieces of iron, and the flat 
sides are securely bolted together. The tube sheet is sufficiently stayed 
by the tubes themselves ; this sheet is often made of copper, as are the 
side sheets, from 6 to 7 inches below, to about the same height above the 
coal line, in locomotive boilers burning anthracite or bituminous coal ; ^ ^ 
are the fire tubes varying in different boilers from li to 3 inches in diam- 
eter and from 7 to 14 feet in length; the longer the tube the larger the 
diameter. H IT are the cylinders, a? a? the steam chests ; uu the exhaust 
pipes, which are connected together, and pass up into the centre of the 
smoke stack or chimney. The exhaust furnishes by its blast draught to 
the chimney, and if the outlet be contracted, the greater the force of the 
issuing steam, and the stronger the draught ; but of course the greater the 
back pressure in the cylinders. 

Plate XL. is the front elevation, and Plate XLI. is the side ele- 
vation and section through air-pump of one of the oscillating engines of the 
Golden Gate, in which a is the main shaft, 1) crank-pin, c cylinder, d trun- 
nions on which the cylinder oscillates to accommodate itself to the motion 
of the crank. <?, stuffing-box on the cylinder head. This is made as long 
as practicable, to give as much bearing as possible for oscillating the 
cylinder, ff^ belt-passage connecting the trunnion with g g^ side pipe. 
li A, valve-stems connecting with the balance puppet-valves, in i i valve- 
chests. Tlio lower valve on the right or steam side is concealed by ^', 
air-pump. The air-pump bucket is provided with India-rnbbcr valves, and 
is worked by Z*, crank on the intermediate shaft. I Z, condenser. There 
are two condensers and two air-pumps, they are located between the cylin- 
ders and inclined towards each other, one only being represeirted. 

Tlie ])assagcyy, together with the side pipes, valve-chests, and appur- 
tenances, are fixed to the cylinder, and oscillate with it, the steam being 
received through one trunnion, and alloAved to escape to the condenser 



PLATE XXXIX. 




I 



FhXI,, 





si 



PTkXIh 




I 



PL.X1,1[. 




I 




X 



\ 




\. 



\ 



N 



\ 




PL.XUIL. 




DRAWIXG OF MACIIINEKY. 205 

tlirougli the opposite one. rn is an injection cock, admitting the water 
upon a scattering plate in the condenser. 

The valves are worked by the toes o o in the nsual manner. Tlie rock- 
shafts ;pjp receive motion partly from the movement of the cylinder, and 
partly from the eccentric. Levers are permanently attached to the trip- 
shafts q^ q^ the ends of which work in a slotted piece curved to the centre 
of the trunnion. Tliis piece is guided, as reju-csented in the engraving, by 
vertical rods sliding in bushes attached to the fixed framing, and is con- 
nected by a rod to the starting-lever r ; all the levers for working by hand 
being so balanced, that the engineer with one hand can work the engine 
TL-p to the usual speed. 

The cut-off valve is placed outside the ti'unnionj and is a balance pup- 
pet-valve, worked by the ordinary cam motion, and so arranged as to act 
either as cut-off or throttle, or both, the levers being placed within reach 
of the engineer when working the engine. 

Plate XLII. is a vertical section through the centre of a Tuebixe 
Wheel, and the axis of the supply pipe." Plate XLIII. is a plan of the 
Turbine and wheelpit. Fig. 1, Plate XLIY., is a plan of the whole wheel, 
the guides and gamitiu-e. This Turbine was constructed for the Tremont 
Manxffacturing. Co. at Lowell, by Mr, James B. Francis, and contains 
most of Mr. Boyden's improvements. Its expenditure of water under 13 
feet head and fall, is about 139 cubic feet per second, and its ratio of useful 
effect to the power expended, about T9 per cent. 

B, the surface of the water in the wlieelj)it, rej^resented at the lowest 
height at wliich the turbine is intended to operate. C, the masonry of the 
wheelpit. D, the floor of the wheelj^it. To resist the great upward pres- 
sure which takes place when the wheelpit is kept diy by pumps, three cast- 
iron beams are placed across the pit, the ends extending about a foot under 
the walls on each side ; on these are laid thick planks, which are firmly 
secured to the cast-iron beams by bolts. To protect the thick planking 
from being worn out by the constant action of the water, they are covered 
with a flooring of one inch boards. E, the wrought-iron supply pipe. 
This is constructed of plate iron three-eighths of an inch thick, riveted 
together. The supply pipe is furnished with the man hole and ventilating 
pipe G, and the leak box H, to catch the leakage of the head gate, when- 
ever it is closed for repairs of the wheel. 

The lower end of the supply pipe is formed by the cast-iron curbs III. 
The curbs are supported from the wheelpit floor by four columns, resting 

* By permission of the author ve take the following plates and description from the standard 
work, " Lowell Hydraulic Experiments." 



206 DRAWING OF MACHINERY. 

on the cast-iron beam, O ; the beams 'N', rest immediately upon the col- 
umns, and the curb npon the beams, the latter projecting over the columns 
far enough for that purpose. The beams 'N' also act as braces from the 
wheelpit wall to the curb, and are strongly bolted at each end. 

K, the disc. This is of cast-iron, and is turned smooth on the upper 
surface, and also on its circumference. It is suspended from the upper 
curb I, by means of the disc pipes M M. The disc carries on its upper 
surface, thirty-three guides (fig. 1, Plate XLIY.,) for the purpose of giving 
the water entering the wheel, proper direction. They are made of Rus- 
sian plate iron, one-tenth of an inch in thickness, secured to the disc by 
tenons, riveted on the under side. The upper corners of the guides, near 
the wheel, are connected by the garniture L, which is intended to diminish 
the contraction of the streams entering the wheel, when the regulating 
gate is fully raised. The garniture is composed of thirty-three pieces of 
cast-iron, carefully fitted to fill the spaces between the guides ; they are 
strongly riveted to the guides and to each other. 

The up23er flange of the disc pipe is furnished with adjusting screws, 
by which the weight is supported upon the upper curb. The escape of 
water between the upper curb and the upper flange of the disc pipe, is 
prevented by a band of leather on the outside, which is retained in its 
place by the wrought-iron ring. The top of the disc pipe, just below 
the upper flange, has two wings, fitting into recesses in the top of the curb, 
to prevent the disc from rotating in the opposite direction to the wheel. 

H, K, the regulating gate. Eepresented, Plate XLII., as fully raised. 
The gate is of cast-iron ; the upper part of the cylinder is stifiTened by a 
rib, to which are attached three brackets, S, S. To these brackets are at- 
tached wrought-iron rods, by which the gate is raised or lowered. To one 
of the rods is attached the rack Y. The other two rods are attached by 
means of links, to the levers T T. Tlie other ends of these levers carry 
geered arch heads, into which, and into the rack Y, work three pinions, 
W, of equal pitch and size, fastened to the same shaft, so arranged that by 
the revolution of the pinion shaft, the gate is moved up or down, cquall}^ 
on all sides. The shaft on which the pinions are fastened, is driven by the 
worm wheel X ; this is driven by the worm a, either by the governor Y, 
or the hand wheel Z. The shaft on which the worm a is fastened, is fur- 
nished with movable couplings, which, when the speed gate is at any inter- 
modiaic ])<)iiils bciwccii its liigliest and lowest positions, are retained in 
place by s])iral springs; in either of the extreme positions, the coni)lings 
are separated by means of a U^ver moved by pins in the rack Y; by this 
means, botli the regulator and hand wheel are prevented from moving the 



DRA^VIXG OF MACHINERY. 207 

gate ill one direction, when tlie gate lias attained either extreme position. 
If, however, the regulator or hand wheel should be moved in the opposite 
direction, the coujDlings would catch, and the gate would be moved. Tlie 
weight of the gate is counterbalanced by weights attached to the levers 
T T, and by the intervention of a lever to the rack Y. 

h h, the wheel, consists of a central plate of cast-iron, and two crowns, 
€ c, of the same material to which the buckets are attached. The buckets 
are forty-four in number, made of Eussian plate iron, /^ of an inch in 
thickness, and are secm-ed to the crowns by grooves cut in the crowns of 
the exact form of tlie buckets, and by tenons entered into the mortises 
in both crowns, and riveted on the opposite sides. 

d d, the vertical shaft, of wrought-iron, runs upon a series of collars, 
resting upon corresponding projections in the suspension box e\ The part 
of the shaft on which the collars are placed, is made separate from the 
main shaft, and is pinned to it at/", by means of a socket in the top of the 
main shaft, which receives a corresponding part of the collar piece. The 
collars are made of cast steel ; they are separately screwed on, and keyed 
to a wrought-iron spindle. 

The suspension box is made in two parts, to admit of its being taken 
off and put on the shaft ; it is lined with Babbit metal. It is found that 
bearings thus lined will carry from fifty to a hundred pounds to the square 
inch, with every appearance of durability. 

f'f', the upper and lower bearings are of cast-iron, lined with Babbit 
metal, adjustable horizontally by means of screws. The suspension box e\ 
rests upon the gimbal g. The gimbal itself is supjDorted on the frame h h 
by adjusting screws, which give the means of raising and lowering the 
suspension box, and with it, the vertical shaft and wheel. Tlie lower end 
of the shaft is fitted with a cast-steel pin, i. Tliis is retained in its place 
by the step, which is made in three parts, and lined with case-hardened 
wrought-iron. 

The weight of the wheel, upright shaft, and bevel geer, is supported 
by means of the suspension box e' on the frame Z,', which rests upon the 
long beams 7/2, reaching across the wheelpit, and supported at the ends by 
the masonry, and also at intermediate points by the braces n n. 

Mr. Francis deduces the following rules for pro23ortioning turbines : 

The sum of the shortest distances between the buckets, should be equal to the diameter 
of the wheel. 

The height of the orifices at the circumference of the wheel, should be equal to one- 
tenth of the diameter of the wheel. 

The width of the crowns should be four times the shortest distance between the buckets. 



208 DKAWING OF MACHINEET. 

Tlie sum of the shortest distances between the curved guides, taken near the wheel, 
should be equal to the interior diameter of the wheel. 

The number of buckets is, to a certain extent, arbitrary. As a guide in practice, to 
be controlled by particular circumstances, and limited to diameters of not less than two 
feet, the number of buckets should be three times the diameter in feet, plus thirty. The 
Tremont Turbine is 8^ feet in diameter, and according to the proposed rule, should have 
fifty-five buckets instead of forty-four. The number of the guides is also to a certain ex- 
tent arbitrary ; the practice at Lowell has been, usually, to have from a half to three- 
fourths of the number of buckets. 

As turbines are generally used, a velocity of the interior circumference of the wheel, 
of about fifty-six per cent, of that due to the fall acting upon the wheel, appears most 
suitable. 

To lay out the curve of the buckets. 

Eeferring to Plate XLIY., fig. 2, the number of buckets, iVJ having been determined 

n D 
by the preceding rules, set off the arc gi= --^ . Let co = ghj the shortest distance 

between the buckets : t the thickness of the metal forming the buckets. Make the arc 
gr ^ = 5a^ Draw the radius OJc, intersecting the interior circumference of the wheel at I; 
the point I will be the inner extremity of the bucket. Draw the directrix I m tangent to 
the inner circumference of the wheel. Draw the arc o n, with the radius a> -\- t, from i, as 
a centre ; the other directrix gp, must be found by trial, the required conditions being, 
that, when the line mils revolved round to the position g t, the point m being constantly 
on the directrix gp^ and another point at the distance mg = rs^ from the extremity of the 
line describing the bucket, being constantly on the directrix m ?, the curve described shall 
just touch the arc no. A convenient line for a first approximation, may be drawn by 
making the angle Ogp== 11°. After determining the directrix according to the preceding 
method, if the angle Ogp should be greater than 12°, or less than 10°, the length of the 
arc g h should be changed, to bring the angle within these limits. 

The trace adopted for the corresponding guides is as follows : — The number n having 
been determined, divide the circle in which the extremities of the guides are found, into n 
equal parts, -y id^ w x, &c. Put co' for the width between two adjoining guides, and t' for 

the thickness of the metal forming the guides. "We have by rule, co' = — . With lo' as 
a centre, and the radius a' -{- t\ draw the arc y z ; and with £C as a centre, and the radius 
2(a)' -\- t'), draw the arc a' h'. Through v draw the portion of a circle i) c', touching the 
arcs y z and a! V ; this will be the curve for the essential part of the guide. The remainder 
of the guide, c' cZ', should be drawn tangent to the curve c' 'c ; a convenient radius is one 
that would cause the curve c' d\ if continued, to pass through the centre 0. 



PLATE XLIV 



•203 




ATvCHITECTUEAL DK AWING. 209 



AECIIITECTUEAL DEAWCTG. 

The art of arcliitectiire consists in the designing of a building, so as to 
be most suitable and convenient for the purposes for which it is intended ; 
in selecting and disposing of the materials of which it is composed, so as to 
withstand securely and permanently the strains and wear to which they 
may be subjected ; and arranging the parts so as to produce the most 
artistic effect consistent with the use of the building, and applying to it 
such appropriate ornament as may express the purpose, and harmonize 
with the construction. 

The art of architecture, according to Ferguson's Hand-Book of ArcJii- 
teeture, should combine the art of the engineer with that of the architect. 
" The art of the engineer consists in selecting the best and most appropriate 
materials for the object he has in view, and using these in the most scien- 
tific manner, so as to insure an economical but satisfactory result. "Wliere 
the engineer leaves off, the art of the architect begins. His object is to 
arrange the materials of the engineer, not so much with regard to econo- 
mical as to artistic effects, and by light and shade, and outline, to produce 
that which in itself shall be permanently beautiful. He then adds orna- 
ment, wdiicli by its meaning doubles the effect of the disposition he has 
just made, and by its elegance throws a charm over the wdiole composi- 
tion. Yiew^ed in this light, it is evident that there are none of the objects 
which are usually delegated to the civil engineer which may not be 
brought wdthin the province of the architect ; a bridge, an aqueduct, a 
pier, are all legitimate subjects for architectural ornament. 

It is not necessary that the engineer should know any thing of architec- 
ture, though it certainly would be better in most instances if he did ; 
but, on the other hand, it is indispensably necessary that the architect 
should understand construction. Without that knowledge he cannot de- 
sign ; but it would be well if, in most cases, he could delegate the mechan- 
ical part of his task to the engineer, and so restrict himself entirely to the 
14 



210 ARCIlITECTUliAL DRAWING. 

artistic arrangement and ornamentation of his design. A building may 
be said to be architectural in the proportion in which the ornamental or 
artistic purposes are allowed to prevail over the mechanical ; and an object 
of engineering, where the utilitarian exigencies of the design are allowed 
to prevail over the artistic. But it is nowhere possible to draw the line 
sharply between the two, nor is it desirable to do so. Architecture can 
never descend too low, nor need it ever be afraid of ornamenting too mean 
objects ; while, on the other hand, good engineering is absolutely indis- 
pensable to a satisfactory architectural effect of any class. The one is the 
prose, the other is the poetry of the art of building." 

Since, then, the basis on which the architect must build is the art of 
the engineer, we continue a former chapter on the Strength of Materials 
with examples of the forms and principles of construction. 

FOUNDATIONS. 

In preparing the foundation for any building, there are two sources of 
failure which must be carefully guarded against : viz., inequality of settle- 
ment, and lateral escape of the supporting material ; and if these radical 
defects can be guarded against, there is scarcely any situation in which a 
good foundation may not be obtained. It is, therefore, important, that 
previous to the commencement of the work, soundings should be taken to 
ascertain the nature of the soil and the lay of the strata, to determine the 
kind of foundation ; and the more important and weighty the superstruc- 
ture, the more careful and deeper the examination. 

Natural foundations. — ^The best foundation is a natural one, such as a 
stratum of rock or compact gravel. If circumstances prevent the work 
being commenced from the same level throughout, the ground must be 
carefully 'benched out^ i. e., cut into horizontal steps, so that tlie courses 
may all be perfectly level. It must also be borne in mind, that all work 
will settle more or less, according to the perfection of the joints, and 
therefore in these cases it is best to bring up the foundations to a uniform 
level with large blocks of stone or with concrete, before commencing the 
superstructure, which would otherwise settle most over the deepest parts, 
on account of the greater number of mortar joints, and thus cause unsightly 
fractures. Foundations in soil should be excavated to a depth below the 
action of frost. 

Artificial foundations. — ^Wherc the ground in its natural state is too 
j=;oft to bear the weight of tlic ju-oposed structure, recourse must be had to 
artificial means of 8iii)port, and, in doing tliis, whatever mode of construe- 



ARCHITECTURAL DRAWING. 



211 




tion be adopted, tlie principle must always be tliat of extending tlie bear- 
ing surface as mucli as possible. There are many ways of doing tliis — as 
by a thick layer of concrete or bdton 
(fig. 1), or by layers of planking, or by 
a net-work of timber, or by increasing 
width of wall, or these different methods 
may be combined. The weight may 
also be distributed over the entire area 
of the foundation by inverted arches. 

The use of timber is objectionable U'"'''''''!'""!""*"'""""""" '"""""""""' ''-"'^'""""'''"i 

where it cannot be kept constantly wet, ^^«- ^• 

as alternations of dryness and moisture soon cause it to rot, and for such 

localities concrete is to be preferred. 

In the case of a foundation partly natural and partly artificial, the ut- 
most care and circumspection are required to avoid fractures in the super- 
structure ; and it cannot be too strongly impressed that it is not an 
unyielding^ but a uniform yielding foundation that is required, and that 
it is not the amount so much as the inequality of settlement that does the 
mischief. 

To prevent the lateral escape of the supporting material, when build- 
ing in running sand or soft clay, which would ooze out from below the 
work and allow the superstructure to sink, in addition to protecting the 
surface with planking concrete or timber, it is often necessary to enclose 
the whole area of the foundation with piles of timber or plank driven close 
together ; this is called sheet-filing 

Where there is a hard stratum below the soft ground, but at too great 
a depth to allow of the solid work being brought up from it without greater 



o o 

o o 

o o 

o o 

o o 

o o 




/ ; 



^;) ^ ^ 



Th ^ t^ \ 



m 



O D.t 



m. ■^^. m 




Fis. 2. 



expense than the circumstances of the case will allow, it is usual to drive 
down wooden piles (fig. 2), often shod with iron, until their bottoms are 



212 ARCHITECTUKAL DRAWING. 

firmly fixed in the hard ground. The upper ends of the piles are 
then cut off level, and covered with a platform of timber, on which the 
work is built in the usual way. The piles are generally of about 1 foot 
diameter, and are driven at distances of from 2 to 3 feet from centre to 
centre. 

Where a firm foundation is required to be formed in a situation where 
no firm bottom can be found within an available depth, piles are driven, 
to consolidate the mass, a few feet apart over the whole area of the foun- 
dation, which is surrounded by a row of sheet-piling to prevent the escape 
of the soil ; the space between the pile-heads is then filled to the depth of 
several feet with stones or concrete, and the whole is covered with a tim- 
ber platform on which to commence the solid work. 

Foundations beneath the surface of water, as for the foundations of 
piers and abutments of bridges, are formed by piles, by throwing down 
masses of stone until the mass reaches the surface of the water, by caisson, 
or by enclosing the space within a coffer-dam, and proceeding as in 
common foundations. 

A caisson is a chest of timber which is floated over the site of the work, 
and being kept in its place, is loaded with stone until it rests firmly on 
the ground. In some cases the stone is merely thrown in, the regular 
masonry commencing with, tlie top of the caisson, which is sunk a little 
below the level of low w^ater, so that the whole wood-work may be always 
covered, and the caisson remains as part of the structure. In others the 
masonry is built on the bottom of the caisson, and when the work reaches 
tlie level of the water, the sides of the caisson are removed. 

Foundations under water are frequently executed with blocks of beton 
or hydraulic concrete, which has the property of setting under water. The 
site of the work is first enclosed with a TO^v of sheet-piling, which protects 
the bdton from disturbance until it has set. The French engineers have 
used beton in the works at Algiers, in large blocks of 324 cubic feet, which 
were floated out and allowed to drop into their places from slings. Tliis 
method, whicli proved perfectly successful, was adopted in consequence of 
the smaller blocks first used being displaced and destroyed by the force 
of the sea. 

WALLS. 

Tlic requisite precaution having been taken to secure a safe foundation 
for tlic Btructurc, tlie next subject to be taken into consideration is the sub- 
stance and pr()j)ortionH of the wallti. 



ARCHITECTURAL DRAWING. 



213 



Walls of 2:)ermanciit structures are almost exclusively composed of 
either stone or brick, or botli, and arc included in one general term as 
masonry. 

Fig. 3 represents tlie front of a wall called the face ; fig. 4, a section ; 
and fig. 5, the view of rear, or the hacking. The interior of the wall is 



^5 



HI 



xzEzr 



^^ 




a. 



dDCDOcjcDa; 



LJ LJI UU 









JZll 



Tig. 3. 



Fig. 4. 



called the filling. The term course is applied to each horizontal layer of 
stone or brick ; if all the stones in a layer are of equal thickness, it is 
termed regular coursing. Footings are the lower projecting courses (figs. 
3 and 4) resting on the foundation, usually not less than double the width 
of the wall above in walls of buildings ; but for other walls, the width de- 
pends on the nature of the foundation. Rock foundations need no extra 
width of wall. String courses, or helting., are upper courses projecting 
slightly beyond the face of the wall. Coping is the top courses, usually 
got out in considerable lengths in comparison with the stones in the rest 
of the work. 

The hed of a stone or brick is the bottom surface on which it rests ; the 
huild is the upper surface on which the stone above is placed ; the inter- 
stices between the stones are termed joints. Stretchers are stones or brick 
which have their length disposed lengthways of the wall ; headers have 
their length crossways. Quoins are the corners of a wall. Bond is the 
lapping of the stones or brick on each other in the construction, so as to 
tie the separate pieces together. Three classes of bond are shown in the 
face (fig. 3) ; the lowest six courses consist of alternate coni'ses of headers 
and stretchers, the next six courses above have alternate headers and 
stretchers in the same course, and in the remaining courses a header 
occurs at every third stone ; this is the most usual bond. Headers should 
not be placed one above the other in alternate courses. 



214 



ARCHITECTURAL DRAWING. 



Figs. 6 and Y represent brick bonds ; — ^fig. 6, the old English bond, and 




l§|\V 


\\\V 


vW 


iS^ II 1 i 1! i 11 i i 




H 11 


11 


^'% 1 " |— IT U i il 11 I— i 




i II 


« 1 



Tig. 6. 



Fig. 7. 



fig. 7, tbe Flemish bond. The most common bond in this country is to 
lay a certain immber of stretching courses, and then a heading course. 
Tlie fire-law of 'New York requires brick work to be built with headers 
every five courses, but every seventh course a heading course is more com- 
monly used. In all masonry, no vertical joint should extend through two 
courses, but the vertical joints should be as near as possible midway be- 
tween those below ; in other words, hreak joint with them. 

Walls are composed of stones laid either with or without mortar. The lat- 
ter is called dry masonry ; rough wall is dry work of rough stones ; if laid 
in mortar, it is called rubble work, but frequently this term is made to include 
all rough work. Cut stone is called ashler ; thus (figs. 3, 4, 5), the face is 
ashler, the backing rubble. Bubble may be either coursed or uncoursed. 

On the thickness of walls. — J^etaining-waih are such as sustain a lateral 

pressure from an embankment 



or a head of water (figs. 8 and 

9). The width of a retaining 

wall depends upon the height 

of the embankment which it 

may have to sustain, and the 

kind of earth of which it is 

composed, (the steeper the 

natural slope at which the 

earth would stand, the less 

the thrust against the wall,) and the comparative w^eight of the eartli and 

of the masonry. The formula given by Morin for ordinary earths and 

masonry kh = 0.285 h + h' ; that is, to find the breadth of a wall laid 

in mortar, multiply the whole height of the embankment above the footing 

285 
by sjr'. ; for dry walls make the thickness one-fourth more. 

Most retaining or brick walls have an inclination or hatter to the face, 
Hometiiiics also the same in the back, but offsets (fig. 8) are more common. 
The usual batter is from 1 to 3 inches horizontal for cacli foot vertical. To 
detcrmiue the thickness of a wall having a batter, " determine the width 




215 



PLATE XLY 



Fi?. 4. 



Fig. 3. Fig. 2. 



r ^1 



V:^ 



Fig. 5. 



^ 



Fig. 1. 



r \ 



Fig G. 



c 1 c :i 



^ 



□ 



Fig. S. 



□ 



^ 



laacjaL 



X 



!□ 



□□tiarziac 



7^—, 



,:^:-r-^ /: :::-A A^" ^A 



ARCHITECTUIIAL DRAWING. 



215 



bj our former rule, aud make this the ^vidtli at one-uintli of the height 
above the base." — Morin. 

The footings or trench walls, as thej are often termed, of a retaining- 
wall should be sunk below the action of the frost, and are generally about 
one foot wider than the wall, offsetting 6 inches both face and back. 

Walls of luildings. — Figs. 1, 2, 3, 4, pi. XLY. represent the thickness 
of external brick walls to the first, second, third, and foiirth-rate buildings, 
as provided bj the Building Act for the city of London. Figs. 5, 6, T, 8, 
show the same with respect to party-walls. The figures 1, IJ, 2, 2J, re- 
present the number of lengths of brick in the wall. The following table 
gives the way in which buildings are rated in Liverpool, not differing, con- 
structively, materially from that of London. 



HEIGHTS AND WIDTHS OF BUILDINGS. 



First-rate dwelling-honse. 



Exceeding forty-four feet 
in height, or twenty-seven 
feet front. 



Second-rate dwelling-house. 



Not exceeding forty-four 
feet in height, or twenty- 
seven feet front. 



Third-rate dwelling-honse. 



Not exceeding thirty-six 
feet in height, or twenty- 
one feet front 



Foarth-rat« dwelling-honse. 



Not exceeding thirty-two 
feet in height, or fifteen feet 
iront 



'' Every brewery, distillery, manufactory, or warehouse, of whatever 
height or extent of frontage, is considered to be a first-rate building, the 
external walls of which are in their respective stories to be 2J, 2, and 1^ 
bricks in thickness, and the party-walls of 2 and 1 J bricks. 

" "Wlien the foundations of any building shall not be upon rock, such 
foundations to have footinsr courses under the same." 



n 



n. 



i_r 



u 



Fig. 10. 



n 



n 



"LT 



U 



^ 



n 



u 



n 



u 



n 






Fig. 11. 



Fig. 12. 




Fi- 13. 



n 



u 






n 



u 



\j 



n. 



_ri 



i_i 



n_ 



Figs. 10, 11, 12, 13, represent the position of the chimneys and thickness 



216 AECHITECTUKAL DRAWING. 

of cliiiniiey-backs in party- walls, according to the several rates; viz., fig. 10 
(first-Yate), basement, 1^ brick, all above, 1 brick thick. Fig. 11 shows 
the thickness when the chimneys are placed back to back in party-walls 
of first-rate buildings. 

Fig. 12 shows the dimensions of second, third, and fonrth-rate build- 
ings, and fig. 13 the same when back to back. 

The following is the Building Act in New York in regard to the thick- 
ness of walls : 

" The outside walls of all dwelling-houses, stores, storehouses, and other 
buildings, shall be not less than eight inches thick ; and all such outside 
walls as shall exceed thirty-five feet in height from the level of the side- 
walk to the peak or highest point thereof, and all party- walls, shall be 
not less than twelve inches thick. And all walls, whether outside, party, 
or partition walls, of any such dwelling-house, store or storehouse, or other 
building other than a dwelling-house, which shall exceed fifty feet in 
height from the level of the sidewalk to the peak or highest point thereof, 
shall not be less than sixteen inches thick to the under side of the second 
tier of beams above the level of the sidewalk : provided said under side 
of said second tier of beams be not less than twenty feet from the level of 
the sidewalk ; but should the under side of said second tier of beams be 
less than twenty feet from the level of the said sidewalk, said walls shall 
be sixteen inches thick to the under side of the third tier of beams above 
the said level. 

'' Every cellar, pier, column, post, and pillar, built of rubble, stone, or 
brick, shall at intervals of not more than three feet have built into it a 
stone of not less than three inches thick, and of a diameter each way equal 
to the diameter of the pier, column, post, or pillar." 

From the above acts we can establish the necessary thickness of walls 
for the usual buildings in a city ; but for detached manufactories and work- 
shops it is usual to adopt a somewhat greater thickness for the walls ; thus, 
for cotton factories of five stories, the first story is generally two feet, the 
second and third twenty inches, and the fourth and fifth 16 inches. Walls 
built of rubble should also be somewhat thicker than those of brick — on 
an average at least one quarter thicker, but depending on the character of 
the stone. Walls of entirely cut stone are stronger than brick of the same 
dimensions, but arc unusual ; the common form of cut-work is to make the 
face ashler, and back up with rubble or brick. 

Jfortar is a mixture of lime or cement, or both, with water and sand. 
In the preparation of good mortar, the materials should be well selected ; 
the sand sharp and clean, the proportions properly preserved, and the 



AECHITECTTJRAL DRAWING. 



217 



whole intimately mixed. As a general rule, the lime or cement should be 
sufficiently line to cover all the grains of sand, and, at the same time, with 
the thinnest possible stratum. Practically, about three to four cubic feet 
of sand are added to one cubic foot of half liquid lime, for fat limes ; lean 
lime may not bear more than half this sand. Cement is generally mixed 
with sand in proportions of one to three, but in situations where a quick 
set is necessary, in equal proportions. 

Arches. — Arches are of various shapes, as. 




7 \:s 


\ 


\ / 

\ / 
\ 

Elliptical, 


_i > 


( - 

^^ / 


Pointed, 


Segmental, 



Circular. 



The outer surface of the arch is called the extrados or laclc of the arch, 
the inner or concave surface the intrados or the soffit ; the joints of all 
arches should be perpendicular to the surface of the soffits. The stones 
are called arch stones or voussoirs, Tlie first course on each side are 
termed springers, which rest on the imposts or abutments. In case of a 
segmental arch, the course beneath the s]3ringers are called sJcew-hacJcs. 
Tlie extreme width between springers is called the span of the arch, and 
the versed sine of the curve of the soffit the ^nse of the arch. The highest 
portion of the arch is called the crown, and the centre course of voussoirs 
the key-course. Tlie side portions of arches between the springing and the 
crown, are termed haunches or flanks. All arches should be well sustained 
by backing on the haunches, called spandrel-hacking. The line of inter- 
section of arches cutting across each other transversely is called a groin^ 
and the arches themselves groined arches. 

Tliat the voussoirs of an arch may resist crushing, they must have a 
certain depth proportioned to the pressure of the arch ; and as this in- 




Fig. 14. 



Fig. 15. 



Fisr. 16. 



creases from the curve towards the springing, the depth of the voussoirs 
should likewise increase from the crown to the springing. Peronnet has 



218 



AECHITECTUEAL DEAWING. 



given as a rule for the depth at the crown the formula cZ = .OT r + 1 foot, 
in which formula t is the greatest radius of curvature of the intrados. 
This formula is applicable to arches less than fifty feet radius ; but beyond 
this it gives greater dimensions than in ordinary practice. In order to 
facilitate investigations on the stability of arches of the more usual forms, 
M. Petit calculated a series of tables, of which we give the abstract for 
circular arches, as the class occurring most frequently in practice. 





CO-EFFICIENT OF HOEIZONTAL 


THEUST AT THE CKOWN. 




Ratio of 

the Radii 

R 


Fig. 14. 


Fig. 16. 


Fig. 15. 






8 = 4h 


8 =5h 


8 = eh 


8=8.1l 


8 = lOh 


8 = 16h 


r 






T = " 


1=3,625 


r 


1=8.5 


i-- 


1 =3« 


1.50 


0.191 


0.21T 














1.45 


0.16S 


0.192 














1.40 


0.162 


0.169 


0.154 


0.147 


0.147 


0.147 


0.045 




1.35 


0.153 


0.14T 


0.148 


0.130 


0.126 


0.126 


0.124 




1.30 


0.143 


0.143 


0.137 


0.123 


0.106 


0.106 


0.104 




1.25 


0.128 


0.139 


0.126 


0.114 


0.100 


0.086 


0.084 


0.072 


1.20 


0.111 


0.131 


0.110 


0.102 


0.091 


0.070 


0.066 


0.066 


1.15 


0.092 


0.119 


0.091 


0.086 


0.079 


0.063 


0.049 


0.041 


1.10 


0.068 


0.103 


0.067 


0.065 


0.062 


0.052 


0.042 


0.027 


1.05 


0.038 


0.082 


0.038 


0.038 


0.037 


0.034 


0.029 


0.019 



To find the thickness of abutment necessary to support the thrust of 
the arch, multiply the co-efficient found in the table for the particular 
case by 3.8, and the square root of the product multiplied by the radius, 
7", of the intrados, will give the extreme thickness of the abutment. 

Example. — ^What is the horizontal thrust, and what the thickness of 
abutment necessary to support an arch of ten feet span and two feet rise ? 



r = 3.625 



-^ , therefore s = 5A. 

2' 

r = 3.625 X 2 = 7.25 ft. 



By Peronnct's formula, d = O.OT x T.25 + 1 = 1.50. 
E = 7.25 + 1.50 = 8.75 
1.20 



8/r5 

7.25 



Py the lablc against 1.20, uiuler the the column 6 = 5 //, wc find 0.102 
as tlic co-efiicieiit of thrust, 150 lbs. being taken as the average weight of 



AECHITECTTJEAL DRAWING. 219 

a cubic foot of masoniy, tlie absolute thrust per square foot of surface is 



0.102 X 150 X 7.25' = 801: lbs. 



V0.102 X 3.8 X 7.25 = 4.50 feet, tliickness of abutment. 

Tlie formula gives the thickness of abutment, supposing the height infinite ; 
for low abutments the thickness may be reduced for common spans about 
ten per cent. 

The following table gives the dimensions of the arches of a selection 
of bridges of European construction. 



LOCATION. 


Material. 


Form of Arch. 


Span. 


Rise. 


Depth 
at crown. 


Depth 
at spring. 








ft. in. 


ft. in. 


ft. in. 


ft. in. 


Manchester and Birmingham Eailroad, 


Bnck, 


Semicircular, 


18 


9 


1 6 


uniform. 


" 


" 


" 


63 


31 6 


3 


« 


London and Brighton " 


« 


(( 


30 


15 


1 6 


2 3 


Blackwall 


" 


Segmental, 


87 


16 


4H 


uniform. 


Great Western 


" 


Elliptical, 


123 


24 3 


5 


rn 


Orleans and Tours " 


Stone, 


Semicircular, 


27 7 




2 7i 


uniform. 


Stirling Bridge, .... 


« 


Segmental, 


60 


13 6^ 


3 6 


4 6 


Carlisle " . . . . 


« 


Elliptical, 


65 


21 


39 


7 4 


Staines « . . . . 


(( 


Segmental, 


74 


9 3 


2 4 


5 6 


Hutcheson " . . . 


" 


» 


79 


13 6 


3 6 


4 6 


Jena " . . . . 


" 




71 9 


10 9 


5 





For smaller culverts of 15 to 30 feet span, the usual construction is to 
make the arch from 1 foot 6 inches to 2 feet deep. Arches in stone are' 
seldom turned less than 1 foot deep, whatever may be the span ; brick 
arches for less than 10 feet span are generally 8 inches, and this depth 
is required by building acts. 

Fig. 17 represents a section of the Croton aqueduct in an open rock 
cut. The bottom is raised with con- 
crete to the proper height and form 
for the inverted arch of a single 
course of brick. Tlie side walls are 
of stone laid in cement, plastered 
and faced with a single course of 
brick. The arch is semicircular, of 
brick, two courses thick, with span- 
dril backing nearly to the level of 
the crown, and the earth filled in on 

top- Fig. 17. 



-;>^^B 


^^^^^- 


>?C^a 


^^^^^^^^^^^^i> 


Scr^-V^^^^ 


1?^^^^^- . -'\r~-j^^I«.>--^ 






[ ^^^ ^%^-^- 


V ->-"^''!o^ 


%^t>_, ">^,, 





220 



AKCHITECTUEAL DKAWING. 



FRAMING. 

Framing is the art of arranging beams for tlie various purposes to 
whicli tliej are applied in structures. Timber and iron are the only ma- 
terials in common use for frames. 

Wooden beams are usually represented longitudinally both in eleva- 
tion and in section by their outlines merely, or in end vievv^ by rectangles 
with diagonals from opposite corners (fig. 18), and in end section by the 

usual diagonal lines in 
one direction across the 





Tis. 18. 



Fig. 19. 



Fig. 20. 



face. Sometimes in 
more finished drawings, 
or when a distinction is to be marked between different materials, the 
grain of the wood is represented as in figs. 19 and 20, a side and end ele- 
vation of a beam. 

Flooring. — ^The timbers which support the flooring-boards and ceiling 
of a room are called the naked flooring. 

The simplest form of flooring, and the one usually adopted in the con- 
struction of city houses and stores is represented in plan and section, fig. 
21. It consists of a single series of beams or deep joists, reaching from 
wall to wall. As a lateral brace between each set of beams, a system of 

JU ^ 



cv 2^ 



^ 



FifT. 21. 



hrl(h/hig is adopted, of wliicli the best is the hcrrlng-l)onc bridging, formed 



ARCHITECTURAL DRAWING. 



221 




of short pieces of joists about 2x4, crossing each other, and nailed se- 
curely to the toj) and bottoms of the several beams, represented by a and 
h in fig. 21, and whereever a flue occurs, or a stairway or well-hole pre- 
vents one or more joists from resting on the Avail, a trimmer or header^ II, 
is framed across the sj^ace into the outer beams or trimmer-leams T T, and 
the beams cut off or tail-heams are framed into the trimmer. 

Whenever the distances between the walls exceed the length that can 
safely be given to joists in one piece, an intermediate beam or girder, 
running longitudinally, is introduced, into 
which the joists are framed (fig. 22). Yery 
often the joists are merely notched on to 
beams. Flooring is still further varied, by 
framing w^itli girders longitudinally ; beams 
crossways, and framed into or resting on the 
girders ; and joists framed into the beams, 
running the same direction as the girders. 
It is evident, that when the joists are not flush or level with the bottom 
of the beams or girders, either that in the finish the beams will show, 
or that ceiling-joists or furrings will have to be introduced. 

Oil the size of joists.- — Tlie following dimensions, taken in part from 
the Liverpool Building Act, may be considered as safe sizes for ordinary 
constructions, the distances from centre to centre being one foot. 

Joists in floors, clear bearing 

Exceeding 7 feet, and not exceeding 10 feet, to be not less tlian 6x2 inclies. 

" " 6 X 2^ " 

" " 7 X 21 " 

" " 8 X 2\ " 

" " 9 X 2| " 

" " 10 X 2f " 

11 X 3 " 

u 12 X 3 " 

It is to be observed that lumber is seldom sawed to dimensions of frac- 
tions of an inch ; we must therefore adopt a width of an integral inch, and 
proportion the distances from centre to centre, according to the increase or 
decrease of width given to the joists. 

Trimmer beams and headers should be of 
greater width than the other beams, depend- 
ing on the distance of the headers from the wall, 
and the number of tail beams framed into it. 
The !N"ew York Building Act requires all headers 
should be hung in stirrup irons (fig. 23), and not framed in. It also re- 



10 " 






12 


12 " 






141 


14i « 






16 


IG " 






18 


18 " 
20 " 






20 
22 


22 " 






24 



u 



A 



Fif 



222 



AUCHITECTUKAL DRAWING. 



quires all girders to be not less than 10 by 12 inches square, and that the 
posts supporting them, shall be placed at intervals of not more than 10 
feet. 

Floors. — In ]N"ew York it is usual to lay single floors, of tongued and 
grooved boards, but in the Eastern States, double floors are more common ; 
the flrst floor consists of an inferior quality of boards, unmatched, laid 
during the progress of the work as a sort of staging for the carpenter and 
mason, and in finishing, a second course is laid on them of better material, 
generally tongued and grooved, but sometimes only jointed. Ceilings 
should always be furred ; that is, laths should never be nailed directly 
to the joists; the usual furrings are of inch board, two inches wide, 
and twelve inches from centre to centre, nailed across from joist to 
joist. 

Fig. 24. represents a section of a mill floor. The girders or beams, 

generally in pairs with a space 
of about an inch between them, 



are placed at a distance of from 
seven to nine feet from centre 



I 



Fiff. 24. 



On these, a 



to centre, and are of from twelve to sixteen inches in depth 
rough plank floor of from three to four inches thick is laid ; the plank 
are dowelled together, that is, put together with pins or dowels, like a 
barrel head. Above the plank is laid the usual top floor, and beneath a 
sheathing of thin boards. 

For extended bearings and for heavy loads, it is often found necessary 
to truss the girders or beams. Fig. 25 represents a bracing truss of 




Fig. 25, 



wrought iron between a double girder ; often a simple piece of arched iron 
is let into the wood, half on each side, and the beams bolted strongly to- 



m 



f-g----^.....- 



Fig. 2G. 



gcthcr. Fig. 20 represents a truss by suspension ; in this case, the strength 
depends upon the cohesive force of the iron. 



ARCHITECTURAL DRAWING. 



223 



Fire-proof floors. — Fig. 27 represents a section of tlic fire-proof flooring 





Fig. 28. 



Fig. 27. 

constructed by Cooper & Hewitt. The girders or beams are of wrought 
iron, with arches of a single conrse of brick in cement between them, 
resting on their lower flanches. The seven-inch deep beams are placed at 
a distance of from three to five feet from centre to centre ; extreme width 
of span, between side walls, fifteen feet. Strips of plank are fastened 
lengthways at the side or on top of the beams, to receive the floor. Fur- 
rings for the ceiling may be attached crossways to the bottom of the beams, 
or the sofiits of the arches may be plastered without any preparation. 

Fig. 28 represents a section of one of the French systems of fire-proof 
floors. It consists of X 
girders, placed at a dis- 
tance of one metre (39.38 
inches), from centre to 
centre, slightly camhered 
or curved upwards in the centre ; the depth of the girders to depend 
upon the span. Stirrups of cast iron are slid upon the girders, into which 
the ends of flat iron joists, set edgeways, pass and are secured by pins; 
the ends of the joists take a bearing also on the bottom flanges of the 
girders. The joists are placed at a distance of one metre from centre to 
centre. Upon the joists rest rods of square iron, which in this way form a 
grillage for the support of a species of rough cast and the ceiling. By 
this and other very similar systems, the French have succeeded in reducing 
the cost of such floors to that of w^ooden ones. 

Floors are sometimes constructed of brick in single or in groined 
arches, the thrust being opposed by the weight of the abutments, but 
owing to its expensiveness, and the amount of room occupied by the ma- 
terial, this kind of construction is not at present very common in edifices. 

Partitions are usually simply studs set at intervals of twelve or six- 
teen inches, these spaces being adapted to the length of the lath (forty- 
eight inches). The sizes of the studs are generally 2 x 4, 3 x 5, or 3 x 6 
inches, according to the height of the partition ; for any high partitions, 
greater depth may be required for the studs, but three inches will be suf- 
ficient width. Partitions should be bridged like floors with herring-bone 
brid2:ino:. 



224: 



AUCHITECTIJKAL DEAWING. 



iJ_ 



UL 



Al—B-ll 



IL 



in I II ii-n ir -ii 



^^ 



VWl 



I n i M 



II II 1 1 II ii 1 1 ir 



A 



r^ 



B 



TTTTT 



F 






Fig. 29. 



Fig. 29 represents the frame of tlie side of a wooden house, in which 

A A are the ^osts, B 
the plate, C C girts or 
interties, D D hraces, E 
sill, F window posts or 
studs, G G studs. 

Usual dimensions of 
timber for frame of com- 
mon dwelling houses : — 
sills 6x8, posts 4x8, 
studs 2 X 4 or 3 X 4, 
girts 6 X the depth of 
floor joists, plates 4 x 
6, the floor joists (J fig. 
30), are notched into the 
girts ; more frequently 
the girts are omitted. 
The studs are of the same length 
as the posts, and the floor joists are 
supported by a board, a, 3 or 4 x 1 
let into the studs (fig. 31), and firmly 
nailed ; the joists are also nailed 
strongly to the studs. Tlie posts and 
studs are tenoned into the sills and 
girts. Fig. 32 represents a tenon, 
1) c, in side and end elevation, and 
onortice, «/ the portions of the end 
of the stud resting on the beam 
are called the sJioidders of the 
tenon. 

Hoofs. — ^The roofs of city dwellings and stores are gencrall}^ flat, that 
is, with but very little inclination, from half an inch to two inches per foot, 
merely sufficient to discharge the water. The beams are laid from wall to 
wall, tlie same as floor timbers, but usually of less depth, or at greater dis- 
tances l)ctwccn centres, and witli one or two rows of bridging. The roof 
is laid witli longued and grooved boards, and mostly covered with tin. 

Figs. J, 2, 3, PI. XLVI. represent side or ])ortions of side elevations of 
tlie usual form of fi-anuul roofs. The same letters refer to the same parts 
in all the iigurcs of the plale. T T are the tie beams, Iv 11 the main rafters^ 



Fig. SO. 
I 



Fig. 31. 



Fig. 32. 



PLATE XLYl. 




AKCHITECTUEAL DEAWIXG. 225 

rr tlie jack rafters^ P P the j^^^^^^^iPP the purlines, KK the hing j^osts^ 
Ti h king holts, q q qiieeii holts, both are called susj)ension bolts, C C the 
collar or straining beams, B B hraccs or studs, h h ridge boards, c c cor- 
bels. 

Tlie ^^^VcA of the roof is the inclination of the rafters, and is usually 
designated in reference to the span as \, \, |, etc., pitch, that is, the height 
of the ridge above the plate is \, \, |, etc, of the span of the roof at the 
level of the plate. Tlie higher the pitch of the roof, the less the tlirust 
against the side walls, the less likely the snow or vrater to lodge, and con- 
sequently, the tighter the roof. For roofs covered with shingles or slate, 
in this 23ortion of the country, it is not advisable to use less than | jDitcli ; 
above that, the pitch should be adapted to the style of architecture 
adopted. The pitch in most common use is \ the span. 

Fig. 1 rejDresents the simplest framed roof; it consists of rafters, resting 
upon a plate framed into the ceiling beam ; this beam is supported by a 
suspension-rod, Ic, from the ridge, but if supported from below, this rod 
may be omitted. This fomi of construction is sufficient for any roof of 
less than 25 feet sjDan, and of the usual pitch, and may be used for a 40 
feet span by increasing the depth of the rafters to 12 inches ; deep rafters 
should always be bridged. By the introduction of a purline extending 
beneath the centre of the rafter, supported by a brace to the foot of the 
suspension rod, as showia in dotted line, the depth of the rafters may ob- 
viously be reduced. It often happens that the king-bolt may interfere with 
the occupancy of the attic ; in that case the beam is otherwise supported. 
Again, it may be necessary that the tie beam, which is also a ceiling and 
floor beam, should be below the plate some 2 to 4 feet ; in that case, the 
thrust of the roof is resisted {^g. 4) by bolts, h h, passing through the 
plate and the beam, and by a collar plank, C, spiked on the sides of the 
rafters, high enough above the beam to aflord good head room. For roofs 
f pitch and under 20 feet span, the bolts are unnecessary, the collar alone 
being sufficient. 

Fig. 2 represents a roof, a larger span than fig. 1 ; the frame may be 
made very strong and safe for roofs of 60 feet sj)an. King-bolts or sus- 
pension-rods are now oftener used than posts, with a small triangular 
block of hard wood or iron, at the foot of the bolts, for the support of the 
braces. The objection to this form of roof is that the framing occupies 
all the space in the attic ; on this account the form, fig. 3, is preferred for 
roofs of the same span, and is also applicable to roofs of at least 75 feet 
span, by the addition of a brace to the rafter from the foot of the queen- 
bolt. The collar beam (fig. 6), is also trussed by the framing similar to 
15 




226 AKCHITECTTJEAL DRAWING. 

In the older roofs, queen posts are used (fig. 33), witb. the foot 
secured by straps or joint bolts to the tie beam. 

In many church and barn roofs the tie beam is cut off, 
fig. 5 ; the queen post being supported on a post, or itself 
extending to the base, with a short tie rod framed into it 
from the plate. 

Figs. 7 and 8 are representations of the feet of rafters 
on an enlarged scale. In ^g. T, the end of the rafter does not 
project beyond the face of the plate ; the coving is formed 
by a small triangular, or any desirable form of plank, 
Fig. 33. framed into the plate. The form given to the foot of the 

rafter is called a crowfoot. In fig. 8, the rafter itself projects beyond the 
plate to form the coving. Fig. 9 represents a front and side elevation and 
plan of the foot of a main rafter, showing the form of tenon, in this case 
double ; a bolt passing through the rafter and beam retains the foot of the 
former in its place. Fig. 10 represents the side elevation of the foot of a 
main rafter with only a small portion of the beam, the remainder being 
supplied by a rod. In fig. 7, of a similar construction to fig. 1, the tie rod 
passes directly through the plate. In general, when neither ceiling nor 
flooring is supported by the tie beam, a rod is preferable. 

Roofs are now very neatly and strongly framed by the introduction of 
cast-iron shoes and abutting plates for the ends of the braces and rafters. 
Fig. 11 represents the elevation and plan of a cast-iron king head for a 
roof similar to fig. 2. Fig. 12, that of the brace shoe ; fig. 13, that of the 
rafter shoe for the same roof. Fig. 14, the front and side elevation of the 
queen head of roof similar to fig. 3, and fig. 15, 
elevation and plan of queen brace shoe. 

Fig. 34 represents the section of a rafter shoe 
for a tie rod ; the side flanches are shown in dotted 
Fig. 84. line. 

On t7ie size and the jprojportions of the different members of a roof: — 
Tic beams are usually intended for a double 2)urpose, and are therefore af- 
fected by two strains ; one in the direction of their length from the thrust 
of the rafters, the other a cross strain, from the weight of the floor and 
ceilin^!;. In estimatint!: the size necessarv for the beam the thrust need not 
l)c considered, because it is always abundantly strong to resist this strain, 
and 1h(! dinicnsions are to l)e determined as for a floor beam merely, 
each ])(»int (if suspension being a sup])ort. Wlien lie rods are used, the 
strain is in tlic dii-t"cli<>ii of ihoir longtlgant)', and tlieir dimensions can bo 
calculated, knowing tlie ])ilc-li, spaumW^weight of the roof 2)er square 





AECHITECTUKAL DRAWING. 227 

foot, and tlie distance apart of the ties, or the amount of surface retained 
by each tie. 

Rule. — Multiply one lialf the weight by one half the span, and divide 
the product by the pitch. 

Example. — What is tlie strain upon the tie rod of a roof 40 feet span 
and 15 feet pitch ? 

The weight of the wood-work of a roof may bo estimated at 35 lbs. 
per cubic foot, or on an average at about 9 lbs. per foot square, slate at 7 to 
9 lbs., shingles at 1| to 2 lbs. The force of the wind may be assumed at 
15 lbs. per square foot. The excess of strength in the timbers of the roof 
as allowed in all calculations, will be sufficient for any accidental and 
transient force beyond this. If, therefore, the roof be like fig. 1, Plate II., 
without ceiling beneath, and retained by a tie rod, we may consider as 
the weight per square foot for a slate roof: 9 + 7 + 15 = 31 lbs. Tlie 
length of the rafter is V20^ + 15" = 25 feet ; hence, if the tie rods are 10 
feet apart, the amount of surface on each incline supported by the tie is 
25 X 10 = 250 square feet, which multiplied by the weight per square 

foot, or 250 X 31 == 7750 lbs. ; applying the rule, -^ x 20 -^ 15 = 5166 

lbs., the thrust on the tie rod. If we estimate the strength of wrought- 
iron at 10,000 lbs. per square inch of section, or 8,000 lbs. when a thread 

is cut upon the end, then, - — - = 0.646 square inches, or a rod a little 

exceeding f of an inch in diameter. 

Tlie rafters (fig. 1, PL XLyi.)maybe considered as jack rafters of long 
bearings, or as a beam supporting transversely the weight of the roof, and 
the accidental pressures, and may be estimated by resolving the direction 
of these pressures into a line perpendicular to the direction of the rafters. 

Main rafters, as in figs. 2 and 3. The pressure on main rafters is in 
the direction of their length, when they are supported by braces at or very 
near the points where the purlines rest ; but in addition to the weight of 
the roof, they support a portion of the weight of the tie beam, and what- 
ever may be dependent upon it. If the frame is like fig. 2, that is, with 
a king-bolt or post, and the weight is uniformly distributed upon the beam, 
then one half the weight is supported by the bolt or post, and consequently 
by the rafter, and the other half by the side walls. Under the same cir- 
cumstances, the suspension rods (fig. 3), support each ^ of the weight of 
the beam, &c., and the side walls each J-. But in general, where the attic 
is made use of, the load is not uniformly distributed, by far the greatest 
part is suspended upon the rods. 



228 AECHITECTUEAL DE AWING. 

To find the pressure on tlie main rafter. Multiply one half the weight 
of the roof, and that portion of the weight of the beam and its load which 
may depend upon it, by the length of the rafter, and divide the product 
by the pitch. 

Example. — What is the pressure upon the main rafter of a slate roof 
of 56 feet span, 21 feet pitch, frames 10 feet between centres, and form 
like fig. 3, with an uniformly distributed load on the beam of 8,400 lbs., 
and a load between the suspension rods of 10,000 lbs. 



The length of rafter is V28' + 21^ = 35 feet. 

Assuming the load per square foot upon the rafter the same as the pre- 
ceding example, 

then 35 X 31 X 10 = 10,850 lbs., J the weight of roof. 

I of the uniform weight — - — •= 2,800 lbs. 

J the weight between rods — - — == 5,000 lbs. 



Total, 18,650 lbs. 
18650 X 35 



21 



31,083 lbs. 



If we now assume the resistance of wood at Y40 lbs. per square inch,^ or 
600 lbs. for length exceeding 13 times their thickness, 

— = 52.59 square inches of section. 

600 ^ 

The proportion of the depth to the width is generally about 10 to 8 or 5 
to 4. 

Hence aJ— x 5 = 1.62 x 5 = 8.1 inches = depth. 

1.62 X 4 = 6.5 inches = width. 



Gwilt, in his Architecture, recommends the following dimensions for 
portions of a roof : 

* Weisbach. 



ABCHITECTUEAL DRAWING. 



229 



•L 



SPAN. 


FORM OF ROOF. 


RAFl'EUS. 


BRACES. 


POSTS, 


coll'r beams. 


feet. 




inches. 


inches. 


inches. 


inches. 


25 


Fig. 2, Plate XLVI. 


5x4 


5x3 


5x5 




30 


u a 


6x4 


6x3 


6x6 




35 


Fig. 3, " 


5x4 


4x2 


4x4 


7x4 


45 


(( u 


6x5 


5x3 


6x6 


7x6 


50 


2 Sets of Queen Posts 


8x6 


5x3 


( 8x8) 
(8x4) 


9x6 


60 


u << 


8x8 


6x3 


j 10 X 8 
< 10 X 4 


11 X 6 



These dimensions, for rafters, are somewliat less tlian tlie usual practice 
in this country ; no calculations seem to have been made for using the attic. 
An average of common roofs here, would give the following dimensions 
nearly : 30 feet span, 8x5 inches ; 40 feet, 9x6; 50 feet, 10 x 7 ; 60 
feet, 11 X 8 ; collar beams the same size as main rafters. Eoof frames 
from 8 to 12 feet from centre to centre. 

Dimensions for jack rafters 15 to 18 inches apart. 



f 6 feet, 


3 X 2i i 


ncli 


8 » 


4x3 


(( 


10 « 


5x3 


11 


12 " 


6x3 


it 


20 " 


10 x 3 


11 



Purlines : 



LENGTH OB' BEARING. 




DISTANCES 


APART IN FEET. 






feet. 


6 


8 




10 




12 


H 


6 X 3i 


6ix 4 




7x5 


8 


X 5 


8 


7x5 


8x5 




9x5 


9 


X 6 


10 


9x5 


10 X 5 




10 X 6 


11 


X 6 


12 


10 x 6 


11 x 6 




12 X 7 


13 


X 8 



The pressure on the plates is transverse from the thrust of the rafters, 
but in all forms except fig. 1, owing to the notching of the rafters on the 
purlines, this pressure is inconsiderable. The usual size of plates for figs. 
1 and 2, is 6 X 6 inches. For forms, fig. 1, the size depends on the mag- 
nitude of roof, and the distance between the ties. The width in all such 
cases to be greater than the depth ; 4 to 6 inches may be taken as the 
depth, 8 to 12 for the width. 

Joints. — As timber cannot always be obtained of sufficient length for 
the different portions of a frame, it is often necessary to unite two or more 



230 



AKCHITECTUKAL DRAWING. 



pieces together by the ends, called scarfing or lapping. Figs. 35 and 36 
are the most common means of lapping or halving, 
_j which methods may be employed when there is 
not much longitudinal compression or extension. 

1 When such an effect is to be provided for, the 

upper as well as the lower timber should be let 
Fig. 36. into each other. Figs. 37, 38, 39, 40, 41, are differ- 



Fig. 85. 



/I 




l^H 



J~2 



Fig. 37. 



Fig. 38. 





ts 



Fig. 40. Fig. 41. 

ent methods to obtain this result. In figs. 37, 38, the joint is brought to a 
bearing by a key driven in tight. Fig. 39 represents a scarf suited for a 
beam supported at this joint by a post, and where there is tensile strain, 
the timber sh(3>uld be joint-bolted or anchored. The centre of the post 
should be beneath the extreme edge of the lower joint. Figs. 40 and 41 
are long scarfs, in which the parts are bolted through and strapped, suited 
for tie beams. Joints are also often made by abutting the j^ieces together 
and bolting splicing pieces on each side ; still further security is given by 
cutting grooves in both timbers and pieces and driving in keys. 

Circular Hoofs. — The rafters of cylindrical roofs may be constructed 
of strips of boards cut to the width of the rafter, bent into the' form of the 
proposed arc, and nailed together to the depth required for the size of the 
roof. The plate extends the whole length of the building ; the foot of the 
rafters arc nailed or bolted to the plate, and tics, cither of wood or iron, 
as often as may be necessary, retain the thrust. For a 50 feet span, of G 
feet rise, the rafters may be 10 x 4 inches, and 15 inches apart; such a 
roof being \^v\i flut on top, nmst be covered with metal. Yery large 
cylin(h-ical roofs are constructed of trusses, those of Howe's bridge (fig. 52), 
the suspension rod being radial, and the lower and upper chtn-ds concen- 
tric ; sucli roofs are nc^arly semicircular. Tlie ends of each truss arc retained 
by tic bolts, and ])iirlines arc used to support the rafters; but in general 



231 



PLATE XLYII. 




AECHITECTURAL DRAWING. 231 

such roofs are for railroad station houses, aud no rafters are used, corrugated 
iron being placed from purline to purline. Eoofs of less than 30 feet span 
are often made of corrugated iron alone, curved into a suitable arc, and tied 
by bolts passing through the iron about 2 to 4 feet above the eaves. 

Iron Boofs. — PL XLYII. fig. 1, represents the half elevation of an iron 
roof of a forge at Paris ; figs. 2, 3, 4, details on a larger scale. Tliis is a 
common form of iron roof, consisting of main rafters, E, of the X section, 
^g. 4, trussed bj a suspension rod, and tied by another ]'od. The purlines 
are also of J iron, secured to the rafters by pieces of angle iron on each 
side ; and the roof is covered with either plate iron resting on jack rafters, 
or corrugated iron extending from purline to purline. The rafter shoe, A, 
and the strut, S, are of cast iron, all the other portions of the roof are of 
wrought iron. 

The surface covered by this particular roof, is 53 metres (164 feet) long, 
and 30 metres (9SJ feet) wide. There are 11 frames, including the two at 
the ends, which form the gables. 

The following are the details of the dimensions and weights of the dif- 
ferent parts : 

lbs. 

2 rafters, 0.72 feet deep, length together, 99.1 feet, . . . 1,751 

5 rods, 0.13 feet diameter, " " 131.4 "... 882 

16 bolts, " 79 

8 bridle-straps, 0.21 x .05 ..... . 123 

2 pieces, .016 thick, connecting the rafters at the ridge, "> 

V . . . 88 

4 " " at the foot of the strut . . > 

4 " .036 thick, uniting the rafters at the junction in the strut — together 

■with their bolts and nuts, . . . . . . 1 70 

2 cast iron struts, ....... 308 

2 rafter shoes, . . ..... 287 

Total of one frame, ....... 3,695 

16 purhnes, 1 ridge iron, each 0.46 deep, 17.2 long, .... 2,985 

Bolts for the same, ....... 64 

16 jack rafters, I iron, 0.16 deep, ...... 2,489 

Weight of iron covering, including laps, per square foot, , . 2.88 

The weight of iron in this roof could be reduced by substituting corru- 
gated iron for the covering, even of less weight per square foot, and omit- 
ting the jack rafters. 

Roofs are sometimes made with deep corrugated main rafters with flat 
iron between, or purlines and corrugated iron for the covering. The great 
objection to iron roofs, lies in the condensation of the interior air by the 
outer cold, or, as it is termed, sweating ; on this account they are seldom 



232 ARCHITECTURAL DRAWING. 

used for other buildings tlian boiler houses or depots, except a ceiling be 
made below to prevent the contact of the air inside with the iron. 

Fig. 5 is an elevation of nearly two of the three panels of one of the 
cast iron girders for connecting the columns, and carrying the transverse 
main gutters, which supported the roof of the English Crystal Palace. 
Figs. 6, 7, 8, 9, 10, 11, sections of various parts on an enlarged scale. 

The depth of the girder was 3 feet, and its length was 23 ft. 3| inches. 
Tlie sectional area of the bottom rail and flange in the centre (fig. 7), was 
6^ square inches ; the width of both bottom and top rail (fig. 6), was re- 
duced to 3 inches at their extremities. It will be observed that the section 
of the braces and ties are such as to give great stiffness, and the section of 
the braces athh (fig. 8), is greater than at c e (fig. 9) ; the section of the 
tie (fig. 10), is the same as the brace at c <? / they are all formed with a 
draft, that is, with a taper from the centre to the outside of from g^ to j\ 
of an inch on a side, according to the depth of the feather. 

The weight of these girders was about 1,000 lbs., and they were proved 
by a pressure of 9 tons, distributed on the centre panel. 

A second series of girders were made of similar form to fig. 5, but of 
increased dimensions in the section of their parts. Their weight averaged 
about 1,350 lbs., and they were proved, as above, to 15 tons. 

A third series, of increased section of parts, weighed about 2,000 lbs., 
and were proved to 22|- tons. 

Fig. 12 represents an elevation of two of the nine panels of one of the 
wrought iron trusses which carry the lead flat and arched roof across the 
nave of the Crystal Palace. These trusses are 72 feet long and 6 feet deep. 
The top rail G, shown in section fig. 13, consists of two angle irons 4J 
inches deep, 3|- inches wide, and f of an inch thick, with a j^late 9 inches 
wide and | thick, riveted on top. A space of 2 inches is left between the 
angle irons. Tlie angle irons are in five lengths, and are connected by 
eight ^ rivets passing through them, and through a plate or plates intro- 
duced between them. The top plate is in seven lengths, connected by ^ 
inch rivets, through the angle irons, the plate, and a joint })late. The top 
plate is riveted to the angle irons by | inch countersunk rivets, 5 inches 
apart. The bottom rail consists of two flat wrought iron bars, G inches 
deep, with a 2 inch space between them. It is in four lengths, jointed by 
six 1 incli rivets passing througli joint plates x 18 x |f inches on the 
outside, and three plates 17 x | inches. The bars forming the central 
lengths of the bottom rail arc | of an inch thick, and those forming the 
side lengths arc ^ of an incli thick. 

Tlio Olid blaiidanls are of cast iron, 3]- inches wide, 4 inches deep, and 



ARCHrrECTUEAL DRAWING. 233 

1 incli tliick, of a T form of section, secured to the column by six l^ inch 
bolts. Tlie standard is 2 inches thick at top and bottom where it receives 
the rails. Two sockets are formed in the middle, to receive the diagonals 
I and J. I, being exposed to compression, is made of four angle irons, 
2 1 X 2^ X j\ inch, riveted together in pairs with i inch rivets. The 
diagonal J is formed of two bars, 4| x | inch, and is secured at each end 
by a If inch rivet. The ends are thickened by short plates riveted to them 
to make up in a measure the loss of strength from the large rivet hole. 
Tlie diagonal K is formed of two bars 4^ inches deep by 1 inch thick, and 
is fixed at each end by a 2 inch bolt and nut. The other diagonals, being 
exposed to much less strain, are formed of single bars 4|^ x j inch, and 
are secured at each end by a 1 inch rivet. 

The standards B and C consist each of four angle irons, 2 1 x 2| x | 
inch, riveted together in pairs, and the two pairs riveted together with six 
small cast iron distance pieces between them. The next standard, that is, 
the thii'd from each end, but not shown in the drawing, is of cast iron. It 
is of + section, being at the centre 6x6 inches, thickness of metal f to | 
of an inch. The base, which rests upon the base of the bottom rail, is 18 
X 4 inches, and the top is 18 x 3 inches. Triangular projections enter 
the top and bottom rail, where they are secm-ed by 1 inch rivets. In the 
centre is a socket or slot through which pass the two light diagonals. 
The main strength of the truss consists in the top and bottom rails, the 
diagonals I, J, K, the first wrought iron standard, B, and the cast iron, 
standard, D. 

On the General Principles of Bracing, — Let fig. 42 be the elevation 
of a common roof truss, and let 
a weight, W, be placed at the 
foot of one of the suspension 
rods. ]^ow, if the construction 
consisted merely of the rafter ^ 
C B, and the collar-beam C C, rig. 42. 

resting against some fixed point, then the point B would support the whole 
downward pressure of the weight ; but in consequence of the connection 
of the parts of the frame, the pressure must be resolved into components 
in the direction C A and C B, C h will represent the pressure in the 
direction C B, C w the portion of the weight supported at B, C a the 
pressure in the direction C A, and w W the portion of the weight sup- 
ported on A. Tlie same resolution obtains to determine the direction and 
amount of force exerted on a bridge truss of any number of panels, by a 
weight placed at any point of its length (fig. 43.) In either case, the 




9S4- 



AECHITECTUEAI. DEAWESTG. 




Fiff. 43. 



effect of the oblique form C A, upon the angle C, is evidently to force 

c €^ it upwards ; that is, a weight placed at one 

side of the frame has, as in case of the arch, 
a tendency to raise the other side. The effect 
of this upward force is a tension on a por- 
tion of the braces, according to the position of the weight ; but as braces, 
from the manner in which they are usually connected with the frame, are 
not capable of opposing any force of extension, it follows that the only 
resistance is that which is due to the weight of a part of the structure. 

Figs. M and 45 illustrate the results of overloading at single points 
such forms of construction. 




Fi- 44. 





Fig. 45, 



Fi-. 46. 



To remedy this effect, if cotmter braces be introduced, as shown in dot- 
ted lines (fig. 46), the tendency of a weight moving across th^ structure is 
to compress the counters and extend the braces. But since, as we have 
said, braces are not usually framed, esj)ecially in wooden structures, to re- 
sist a tensile strain, it is necessary to overcome this force in another way ; 
that is, by introducing wedges between the end of the counter braces and 
the joints against which they abut, or by means of the counters and the 
suspension rods, in any way straining the structure so that there may be 
an additional compression upon the brace more than the upward or tensile 
force exerted by any passing weight. In this case, therefore, the passage 
of a load would produce no additional strain upon any of the timbers, but 
would tend to relieve the counters. The counter braces do not, of course, 
assist in sustaining the weight of the structure ; on the contrary, the 
greater the weight of the structure itself, the more will the counter braces 
be relieved. 

If instead of the counter braces, the braces themselves are made to act 
both as tic, and as a strut, as has been done sometimes in iron bridges and 
trusses, tlieii tlie upward force will be counteracted by the tension of the 



ARCHITECTURAL DRAWING. 



235 



brace, but in general connter braces are preferable, as it is better tliat tlie 
force exerted against any portion of the structure should always be in one 
direction. 

It follows, from what has been shown of the effect of a variable load, 
that no bridge, either straight or arched, intended for the passage of heavy 
vehicles or trains, should ever be without counter braces or diagonal ties, 
and that only in the case of roofs or aqueducts of similar construction, 
vrhen the load is uniform, or very small in comj)arison with the weight of 
the structure itself. 

On the Truss hy Tension Bod (fig. 4T). — Since the limit of the elas- 




Fig. 4T. 

ticity of iron is very small in comparison with wood, w^hen iron is thus 
used to truss timber, the rods must break before the beam reaches the de- 
flection that the weight should produce. It is evident, therefore, that in 
construction the beam should not be cambered by the tension of the 
rod, but that the top of the beam should be arched, and be permitted to 
settle with the w^eight before it strains the rod at all. In general, the rods 
should, be depended on to resist the whole of the tension, and act as the 
lower chords of an ordinary bridge ; in this way the calculation becomes 
very simple and furnishes safe practical results. Thus (fig. 4T.), to estimate 
the strain upon the suspension rod, multiply the weight supported at the 
point G or c' by the length of the rod a d or d' ^, and divide the product 
by the length of the strut c d. The length of the rod and of the strut, 
may be measured from any horizontal line which completes the triangle. 

Examjple. — What must be the tension on the rod of a truss 40 ft. span, 
supporting a load uniformly distributed of 80,000 lbs. ; the struts c d, c' d\ 
being 3 feet, and the middle interval 12 feet, and the end ones 14 feet 
each? Tlien each point c, c' ^ supports '/ + y ~ 40 or \l oi the weight, 
and the side walls /^ each. 

J- X 80,000 = 26,000, w^eight on c or c' . 



4/14' + 3' = 14.32, length of rod. 
26000 X 14.32 



124106 

8000 



= 124,106 lbs., tension of rod. 
= 15.5 sq. inches necessary to resist this tension. 



236 



AKCHITECTUKAL DRAWING. 



Suppose a system to be composed of a series of suspension trusses, as 




Fiff. 48. 



in 'G.g. 48, in wMcli the load is uniformly distributed. K we represent the 
load at each of the points, 4, 3, 2, 1, 2', &c.,,by 1, the load at 4 will be 
supported | upon a and J upon 3 ; hence the strut 3 will have to support 
a load of 1 + .5 = 1.5 ; of this, | will be supported by 2 and | by «; | 
of 1.5 = 1, 1 + 1 = 2, load on strut 2 ; | of this load, or 1.5, will be supported 
at 1, and since from the opposite side there is an equal force exerted at 1, 
therefore the strut 1 supports 1+1.5 + 1. 5 = 4; the tension on the rod 
c 2 is 4 ; on 2 3, 4 + 1 = 5 ; on 3 4, 5 + 1=6; on 4 «, 6 + 1 = 7 ; and 
the rod should therefore be increased in strength in these proportions from 
the central point c, to the point of suspension, a. The tension on the rods 
3 4, 2 3, 12, may be easily resolved from their direction and the load upon 
the several struts. 

If this construction be reversed, the parts which now act as ties must be 
made as braces, and braces, ties ; then we have a roof truss, and the force 
exerted on the several parts may be estimated in a similar way as for the 
suspension truss. 

It is evident that neither of these constructions would serve for a bridge 
truss, subject to the passage of heavy loads, but is only fit to support uni- 
form and equally distributed loads. 

To frame a construction so that it may be completely braced, that is, 
under the 'action of any arrangement of forces ; the angles must not admit 
of alteration, and consequently the shape cannot. The form should be 
resolvable into either of the following elements :• — ^Figs. 49, 50, 51. 



Fig. 50. Fig. 51. 

. represent parts required to resist 




Fig. 49. 

In tliese figures, lines 
compression ; lines 



parts to resist tension only ; lines 

II >. parts to resist both tension and compression. 
It is evident that in a triangle (fig. 40.), an angle cannot increase or 



ARCniTECTUKAL DRAWING. 



23* 



diminisli, without tlie opj^osite angles also increasing or diminishing. In 
tlie form fig. 50, a diagonal must diminish ; in fig. 51 a diagonal must 
extend, in order that any change of form may take place. Consequently 
all these forms are completely braced, as each does not permit of an effect 
taking j)lace, which would necessarily result from a change of figure. 
Hence also, any system composed of these forms, properly connected, 
breaking joint as it were into each other, must be braced to resist the 
action of forces in any direction ; but as in general all bridge trusses are 
formed merely to resist a downward pressure, the action on the top chord 
being always compression, it is not necessary that these chords should act 
in both capacities. 




Fig. 52. 



Yis. 53. 



We instance figs. 52 and 53, as illustrative of two systems of bridge 
trussing, the one being an elevation of Howe's bridge truss, the other of 
Pratt's. There are many other forms of trussing, but sufficient has been 
said to explain the principles of most constructions. For farther illustra- 
tion, we would refer to Haupt on Bridge Construction. 

DRAWING. 



In the earlier chapters of this work we have given descriptions of the 
usual variety of drawing instruments and their use, and it is to be pre- 
sumed that the reader is sufficiently acquainted with their management to 
construct, with but little explanation, most architectural drawings. 

It has been observed, that the first thing to be done towards the execu- 
tion of a drawing is to determine the scale upon which the drawing is to 
be made. The usual scale for plans and elevations in architectural di*aw- 
ings is either four feet or eight feet to the inch, and especially in all work- 
ing drawings it is necessary that the scale should be such that it can be 
measured with the common two-foot rule, whose divisions are in eighths 
or sixteenths of an inch. Working details are generally drawn on coarse 
paper, and on as large a scale as possible, often full size. In our own illus- 
trations, the size of the page has confined us to scales much smaller than 



238 AKCHITECTUKAL DEAWESTG. 

should be recommended in practice, and the learner, in copying tliem, 
should in no case adopt the same scale, but select one for himself. 

Plate XLYIII. represents the plans, end and side elevation, and section 
of a house. 

To construct these, select a scale of say 4 feet to the inch, and com- 
mence with the plan fig. 1. Lay oif a base line A B on this measure of 
20 feet for the length of the house, and erect perpendiculars at the extrem- 
ities thus measured. Lay off 16 feet for the width on each of the perpen- 
diculars, and connect these points, the line will be parallel to A B, and the 
outline of the house will be defined. The thickness of the wall for a house 
of this size, if of brick, will be 8 inches for wall and 2 inches for furring 
and plastering, or 10 inches ; if of wood, the studs should be 2 x 4 ; 1| 
inches for outside boarding and 1 for lath and plaster, or 6|- in total. Lay off 
now on the inside of the outline the thickness of the wall, and draw the inte- 
rior lines. Lay off now the partitions marking the rooms ; these partitions 
2xiay be represented by a single line or by two lines. The thickness of 
partition for such an edifice will be from 4 to 5 inches. The dimensions, 
as generally marked, should be from outside lines to centres of partitions, 
or from centres to centres of partitions, as much more determinate for the 
carpenter to work from than dimensions in the clear, that is, between par- 
titions, or actual space in the rooms. Lay off now the position of the 
windows, which are to be 3 feet wide. It is to be observed that we must 
strive in general to preserve uniformity both in inside and outside gippear- 
ance. Here we wish the front window to be, as near as possible, in the 
centre of the room, and also, on the outside, to be uniform with the posi- 
tion of the door. The side of the door may be 4 feet from the end of the 
house ; mark its position and width ; but the side of the window, if in the 
centre of the room, will be 5 feet 4 inches from the opposite corner; make 
it 5 feet, and lay off' the window in the rear opjDosite to the front window. 
The openings for windows are distinguished from those for doors by 
straight lines drawn across the aperture. The window in the end of the 
liouse, to present an uniform appearance outside, should be in the centre, 
and this position will also suit the purpose for which the room is designed, 
a blank corner being necessary for the bed. Lay off now the position of 
the fireplace in the centre of the end of the room and of the structure, 
the opening to bo 3 feet, the width of the jambs 8 inches each, and the 
width of the ])ack 2 feet 4 inches. Draw lines for a partition fiusli with 
tlie face of lhc», cliinmcy to one side of the room, witli an opening for a 
door to be for a closet or pantry; all the inside doors to be 2 feet 10 inches 
wide, and to be represented by openings in the partitions nierel}'. Lay 



PLATE XLYIII. 



236 




ARCHITECTURAL DRAWING. 239 

off the stairs, as shown in the lobby, 2 feet 9 inches long. Kow fill in the 
space between the interior and exterior outlines of the walls and partition 
in black, leaving the openings for the doors and windows without color, 
and the plan is complete. The filling is not necessary, but it adds consid- 
erably to the distinctness, and is more ex2)lanatory to persons not conver- 
sant with drawings, than leaving it in plain outline. 

To construct the front elevation (fig. 2), project the various points or 
position of the corners, window and door, as shown on the 23lan, and extend 
the lines of projection as high as may be necessary above the line E D ; 
on these lines mark off the height of window and door, the height of the 
eaves in projecting must be determined from the end elevation; finish the 
w^indow and door w^ith lines to represent caps and sills, and wdiatever other 
lines may be necessary for the style of finish of the drawing. On the line 
C D erect also the outlines of the end elevation, taking the horizontal di- 
mensions from the plan, and projecting, as far as possible, the vertical ones 
from the front elevation. Establish the ridge of the roof by setting off 
the height of the pitch on the centre line above the eaves, and draw the 
various lines to represent the boarding and cornice. Draw the chimney 
in the centre above the ridge. It is to be observed that this chimney is at 
the opposite end of the house, and may be represented in lighter lines than 
the rest of the drawing. The base of the chimney and the eaves are to 
be projected on the front elevation. It is to be observed that many lines 
may be projected, either from the front to the side elevation, or vice versa; 
the true way is to construct both elevations together. 

The outline of the section (fig. 4), may be taken from the end elevation 
or may be constructed directly from the dimensions. The roof is con- 
structed of the plainest form, as being of very moderate sj^an, and the 
attic may be finished for occupation. The dimensions given in this par- 
ticular instance, in all the figures, is such as would be sufficient for the 
class of building and purposes for which it is designed. The learner will 
copy the details as laid down, but on the scale which has been adopted. 

The cellar plan (fig. 5), is easily deduced from the ground or first fioor 
plan (fig. 1). 

PL XLIX. represents plans of familiar forms of houses, all drawn to the 
same scale, as illustrations to the student, and as examples to be copied on 
a larger scale. The same letters of reference are used, on all the plans, for 
rooms intended for similar purposes. Thus, K K designate kitchens, cook- 
ing rooms, or laundries, D D eating rooms, S S sleeping rooms, P P draw- 
ing rooms, parlors, or libraries, jpp pantries, china or store closets, or clothes- 
presses, c G water-closets and bath rooms. 



240 AECHITECTUEAL DRAWING. 

The plans and elevations given in Plate XLYIIl. represent as small a 
dwelling house as there should be occasion for building. Fig. 1, Plate XLIX., 
is the first floor plan of a house containing one room more, which may be 
used either as an eating or drawing room. Pigs. 2 and 2^, are the first and 
second floor plans of a still larger house. Pig. 3 is the floor plan of the 
same house differently arranged, the kitchen being in the basement. Pig. 
4, the same, with an L in the rear for the kitchen. Tliese plans are all of 
square houses, and although not picturesque in their elevations, are yet 
very convenient and economical structures; they are intended for the 
country. The sheds and back offices should be beneath a different roof, 
but attached or not to the main building as may be desired. 

Pigs. 5, 6, and Y, are first floor plans of houses of a different outline, 
but yet imiform, or nearly so ; in subsequent plates will be found illus- 
trations of more varied forms of houses. In the cities, houses are 
mostly confined to one form in their general outline, a rectangle. Pigs. 
8 and 9 may be taken as the usual type of 'New York City houses. Pigs. 
8, 8', 8'', are the basement, first, and second fioor plans of a Basement 
house, three rooms deep. There is usually a cellar beneath the basement, 
but in some cases there are front vaults, entered beneath the steps to the 
front door ; the entrance to the basement itself is also beneath the steps. 
The front room of the basement may be used as an eating room, for the 
servants' sleeping room, billiards, or library. The usual dining room is on 
the first floor ; a dumb waiter being placed in the butler's pantry j>, 
for convenience in transporting dishes to and from the kitchen. The ob- 
jection to three room deep houses is that the central room is too dark, being 
lighted by sash folding doors between that and the front or rear rooms or 
both. Pig. 8^^' is a modification to avoid this objection, the dining room, 
or tea room as it is generally called, being built as L, so that there is at 
least one window in the central room opening directly out-doors. Pigs. 9, 
9', 9'^, 9'^^, are plans of the several floors of an English Basement house so 
called, distinguished from the former in that the principal floor is up one 
flight of stairs. The first story or basement, is but one or two steps above 
the street, and contains the dining room, with its butler's pantry and dumb 
waiter, a small sitting room, with, in some cases, a small bed room in the 
room in the rear of it. The kitchen is situated beneath the dining room, 
in the sub-basement. The grade of the yard is in general some few steps 
above tlic floor of tlic kitchen. Vaults for coal and ])rovisions are exca- 
vated either l)encath tlie pavement in front or beneath the yard. The 
advantages of this form of house arc the small sitting room on the first 
floor, wliich iu small families, and in the winter months, is the most fre- 



PLATE XLIX. 



Fij;. 8. 



u WL. 
^' II 



Ll_ 



Fit:. 9. 



r 



la 



II 



K I 



Fi- 1. 

p I 



K 



R. 



"1 



L'J 



II 






Fi^ 



1 



m 



S 






L X 



jD 



C 



g 



cP 



_j 






t 

P 



Fig. 2. 




jfiT 



r 



/Ft 



L;'_t'r 



Fi- -3. 



Fi- 4. 



i 7^ 



Fir. C. 



LJLj 



\ll!f 



ISii 



i 



L ^ 



M r ' 




AECHITECTURAL DRAWING. 241 

quentlj occui^ied of any in the liouse ; the spaciousness of its dining room 
and parlors in proportion to the width of the house, which is often but 16 
feet 8 inciies in width, or three houses to two lots, and not unfrequently of 
even a less width. The objections to the house are the stairs, which it is 
necessary to traverse in passing from the dining rooms or kitchen to the 
sleeping rooms, but this objection would, of course, lie against any house 
of narrow dimensions, where floor space is supplied by height. 

On the Size and Projyortion of Rooms in general. — *• Proportion and 
oiTiament," according to Ferguson, " are the two most important resources 
at the command of the architect, the former enabling him to constnict or- 
namentally, the latter to ornament his construction." A j)roportion to be 
good, must be modified by every varying exigence of a design, it is of 
course impossible to lay down any general rules which shall hold good in 
all cases ; but a few of its princij)les are obvious enough. To take first 
the simplest form of the proposition, let us suppose a room built, which 
shall be an exact cube — of say 20 feet each way — such a proportion must 
be bad and inartistic ; (and besides,) the height is too great for the other 
dimensions. As a general rule, a square in plan is least pleasing. It is 
always better that one side should be longer than the other, so as to give a 
little variety to the design. Once and a half the width has been often 
recommended, and with every increase of length an increase of height is 
not only allowable, but indispensable. Some such rule as the following 
meets most cases ; " The height of the room ought to be equal to half its 
width plus the square root of its length ; " but if the height exceed the 
width the effect is to make the room look narrow; again, by increasing the 
length we diminish, apparently, the other two dimensions. Tliis, however, 
is merely speaking of plain rooms with plain walls ; it is evident that it will 
be impossible, in any house, to construct all the rooms and passages to 
conform to any one rule of proportion, nor is it necessary, for in many 
rooms it would not add to their convenience, which is often the most desir- 
able end ; and if required, the impleasing dimensions may be counteracted 
by the art of the architect, for it is easy to increase the apparent height by 
strongly marked vertical lines, or bring it down by horizontal ones. Thus 
if the walls of two rooms of the same dimensions be covered with the same 
strongly marked striped paper, in one case the strijDes being vertical, and 
in the other horizontal, the apparent dimensions will be altered very con- 
siderably. So also a deep bold cornice diminishes the apparent height of a 
room. If the room is too long for its other dimensions, this can be reme- 
died by breaks in the walls, by the introduction of pilasters, etc. So also, 

as to the external dimensions of a wall, if the length is too great it is to be 
16 



242 AECHITECTUEAL DRAWING. 

remedied by projections, or by breaking up the lengths into divisions. 
This will be understood by reference to elevations of " Country Houses," 
Plates LXy. to LXIX. In this view, as variety in form adds greatly to the 
picturesque, it is far better in designing a country house, where one is not 
restricted to room, to mark out the rooms to the size which we wish them to 
be, cutting out slips of paper of the dimensions, according to some scale, to 
arrange them then in as convenient an order as possible, and again modifying 
the arrangement by the necessities of construction and economy. Thus 
the more the enclosing surface, in proportion to the included area, the 
greater the number of chimneys, unnecessary extent of passages; all of 
course conduce to an excess of expense. Again, the kitchen should be of 
convenient access to the dining room, both should have large and commo- 
dious pantries, and all rooms should have an access from a passage, with- 
out being compelled to pass through another room ; this is particularly 
applicable to the communication of the kitchen with the front door. Out- 
side doors for common and indiscriminate access should open into passages 
and never into important rooms. 

As to the size of the different rooms, they must of course depend on 
the purposes to which they are to be applied, the class of house, and the 
number of occupants. To commence with the kitchen, for the poorer class 
of houses it is also used as an eating room, and should therefore be of con- 
siderable size to answer both purposes ; for the richer houses, size is neces- 
sary for the convenience of the work ; in 'New York City houses the aver- 
age will be found to be about 16 x 20 feet, for medium houses in the 
country they are in general less, say 12 x 16. A back kitchen, scullery, 
or laundry, should be attached to the kitchen, and may serve as a passage 
way to out of doors. 

The dining or eating rooms. — ^The width of dining tables vary from 3' 
to 5 feet 6 inches, the space occupied by the chair and person sitting at the 
table is about 18 inches on each side ; the table space, for comfort, should 
be not less than 2 feet for each ]3erson at the sides of the table, and 
considerable more at the head and foot ; hence we may calculate the space 
that will be necessary for the family and its visitors, at the table. If we 
now allow a farther space of 2 feet at each side for passages, and some 3 
to 5 at the head for the extra tables or chairs, we can mark out the mini- 
mum of space required : but, if possible, do not confine the dining room to 
meagre limits, unless for very small families ; let not the parties be lost in 
the extent of space, nor let them appear crowded. 

The show room jMrlors, if there arc any intended for such in the house, 
may be made according to the rules given above, not square, but the length 



ARCHITECTURAL DRAWING. 243 

about once and a half the width ; if much longer than this, break up the 
walls bj transoms or projections. As to the particular dimensions no rules 
can be given, it must depend on every person's taste and means. 20 x 16 
may be considered a fair medium size for a regular living room parlor, not 
a drawing room. The same size will answer very w^ell for a sleeping room. 
Tlie usual width of single beds is 2 feet 8 inches, of three-quarter 3 feet 
6 inches, of whole 4 feet 6 inches, the length G feet G inches, so that if 
adequate means of ventilation are provided, it is easy to see into how small 
quarters persons may be thrust. The bed should not stand too near the 
fire, nor between two windows ; its most convenient position is head again-st 
an interior wall, with a space on each side of at least 2 feet. 

Pantries. — Closets for crockery should not be less than 14 inches in. 
width in the clear ; for the hanging up of clothes, not less than 18 inches, 
and should be attached to every bed room. For medium houses, the closets 
of large sleeping rooms should be at least 3 feet wide, with hanging room, 
and drawers and shelves. There should also be blanket closets, for the 
storing of blankets and linen ; these should be accessible from the entries, 
and may be in the attic. Store closets should also be arranged for gro- 
ceries and sweetmeats. 

Passages. — The front entries are usually G feet wide in the clear ; com- 
mon passage ways are usually 3 feet wide ; these are what are required, but 
ample passages give an. important effect to the appearance of the houses. 
Tlie width of principal stairs should be not less than 3 feet, and all first 
class houses, especially those not provided with water-closets and slop 
sinks on the chamber floor, should have two pair of stairs, a front and 
a back pair ; the back stairs may not necessarily be over 2 feet G inches 
in width. 

The Height of Stories. — It is usual to make the height of all the rooms 
on each floor equal, it can be avoided by furring down, or by the break- 
ing up of the stories, by the introduction of a mezzonine or intermediate 
story over the smaller rooms. Both remedies are objectionable ; the more 
artistic way is to obviate the appearance of disproportionate height by 
means stated above. 

The average height of the stories for such city dwellings as we have 
given plans of are : cellar G feet G inches, common basement 8 to 9 feet, 
English basement 9 to 10, principal story 12 to 15, first chamber floor 10 
to 12, other chamber floors 8 to 10 feet, all in the clear. For country 
houses the smaller of the dimensions are more commonly used. Attic 
stories are sometimes but a trifle over G feet in height, but are of course 
objectionable. 



244: AKCmTECTUEAL DEAWING. 

Details of parts. Stairs consist of the tread or step on whicli we set 
our feet, and risers^ upriglit pieces supporting tlie treads — each tread and 
riser forms a stair. If the treads are parallel they are called fliers, if less 
at one end than the other, they are called icinders ; ia figs. 9' and 9'', Plate 
XLIX., both kinds are shown. Any wide step, for the purpose of resting, is 
called a landing. The height from the top of the nearest step to the ceil- 
ing above, is called the headway. The edge of the step which is rounded 
is called a nosing, as fig. 6, Plate L. ; if a small hollow l be glued in 
the angle of the nosing and riser, it is called a moulded nosing. The pieces 
which support the ends of the stairs are called strings, that against the 
wall the wall string, the other the outer string. Besides the strings, 
pieces of timber are framed and placed beneath the fliers, called carriages. 
The opening on plan (which must occm* between the outer strings, if they 
are not perpendicular over each other), is called the ^uell hole, A, fig. 3, 
Plate L. 

The breadth of stairs in general use is from 9 to 12 inches. In the 
best staircases, the breadth should never be less than 12 inches, nor more 
than 15. The height of the riser should be the more, the less the width 
of the tread ; for a 15 inch tread the riser should be 5 inches high, for 12 
inches, 6|-, for 9 inches, 8. In laying out the plan of stairs, having deter- 
mined the starting point either at bottom or top as the case may be, find 
exactly the height of the story; divide this by the height you suppose the 
riser should be. Thus (fig. 1 ), if the height of the story and thickness of 
floor be 9 feet, and we suppose the riser should be about 7 inches high, 

then 108 inches, divided by 7 = 15f . 

It is clear that there must be an even mmiber of steps, either 16 or 15; 
to be near to what we have supposed to be the height of the riser, adopt 15, 

then VV — ^i\ inches, height of riser. 

For this particular case we assume the breadth of the step as 10 inches, 
and the length at 3 feet, a very usual length, seldom exceeding 4 feet in 
the best staircases of private houses. 

Lay off (fig. 1), the outside of the stairs, two parallel lines 3 feet apart, 
and space off from the point of beginning 14 treads of 10 inches each, and 
draw the cross parallel lines, and we have the plan of the stairs. It is to 
be observed that the number of treads is always one less than the number 
of risers, the reason of whicli will appear by observing the elevation, fig. 2. 

To construct the elevation, the line of the stair in \A'A\\ may be pro- 
jected, and the height l)e divided into the number of risers, 15 of 7} inches 
each, and cross parallels drawn through these points. 



PLATE L. 



244 




ARCniTECTUKAL DRAWING. 245 

Wlien the stairs ai'c circular, or consist in part of winders and fliers, as 
in fig. 3, tlie width of the tread of the winders should he measured on the 
central line. Tlie construction of the elevation of a circular flight may be 
understood from figs. 3 and 4. 

The drawing of elevation of stairs is in general necessary, to determine 
the opening necessary to be framed in the upper floor, to secure proper 
headway. Thus (fig. 2), tlie distance between the nearest stair and the 
ceiling, at a^ should not be less than 6 feet ; a more ample space improves 
the looks of the stairway ; but if we are confined in our limits, this will de- 
termine the position of one trimmer, the other will be of com'se at the top 
of the stairs. TThen one flight is placed over another, the space required 
for timber and plastering, imder the steps, is about 6 inches for ordinary 
stairs. 

Fig. 5 represents a circular flight of stairs without a well hole, the nar- 
row ends of the winders being mortised into a central shaft or newel. 
Tlie same term is also applied to the first 'baluster or post of the hand rail. 
The objection to all circular stairs of this form, or with a small well hole, 
is that there is too much difl'erence between the width of the tread, but a 
small portion being of a suitable size. The handsomest and easiest stairs 
are straight runs^ divided into landings, intermediate of the stories, and 
either continuing then in the same line, or tiu-ning at right angles, or 
makino; a full return. 

Tlie top of the hand rail should, in general, be about 2 feet 10 inches 
above the nosing, and should follow the general line of the steps. Tlie 
angles of the head rail should always be eased oflP, as shown at the top and 
bottom in fig. 2. A common form of the hand rail is shown (fig. V) ; to 
serve the purpose for which in part it is designed, that is, of afiPording an 
assistance in ascending or descending, it should not be wider than the 
grasj) of the hand, and where for architectural eflfect a more massive form 
may be necessary, it is very convenient, and may be very ornamental, to 
have a sort of double foiTo, that is, a smaller one planted on top of the 
lai-ger, as in ^g. 8. 

Doors. — Fig. 9 represents the elevation, and fig. 10 the horizontal sec- 
tion of a common inside door. A A are the stiles^ B, C, H, D, the lottom^ 
lock^ parting ^ and top rail^ E the pajiels, and F the miintin ; the combina- 
tion of mouldings and offsets aroimd the door, G, is called the architrave / 
in the section, a a are the partition studs, hh the door jaml/s. 

"With regard to the projDortions of internal doors, they should depend 
in some degree on the size of the apartments; in a small room a large 
door always gives it a diminutive appearance, but doors leading from the 



246 AUCHITECTUEAL DRAWING. 

same entry, wMch. are brought into tlie same yiew, slionld be of uniform 
height. The smaller doors which are found on sale are 2 ft. 4 in. x 6 feet ; 
for water closets, or very small pantries, they are sometimes made as nar- 
row as 20 inches, but any less height than 6 feet will not aiford requisite 
head room. 2 ft. 9 in. x T ft., 3 ft. x T ft. 6 in., or 3 ft. 6 in. x 8 ft., are 
well proportioned 6 panelled doors. But the apparent proportions of a 
door may be varied by the omission of the parting rail, making the door 
4 panelled, or narrowed still more by the omission of the lock rail, making 
a 2 panelled door. Sometimes the muntin is omitted, making but one 
panel ; but this of course will not add to the appearance of width, but the 
reverse. Wide panels are objectionable, as they are apt to shrink from 
the mouldings and crack. 

"When the width of the door exceeds 5 feet, it is generally made in two 
parts, each part being hung to its side of the frame, or one part hung to 
the other, so as to fold back like a shutter ; or the parts may be made to 
slide back into pockets or grooves in the partition, as shown in plan and 
horizontal section, figs. 11 and 12. One of the doors in the drawing is 
shown as a sash door, the other close panels, so as to give two illustrations 
in the same diagram ; the same may be said of the architrave. It may be 
unnecessary to say that in construction both sides and doors should be uni- 
form. The upper panels of the close door may be made of glass ; the 
finish around this half of the door is with an architrave, as in fig. 9, but 
with different mouldings. The finish over the other half of the door is an 
entablature, supported by pilasters A, commonly called by carpenters 
antse, though not correctly so, the antse being pilasters at the end of a pro- 
jecting wall. 

Figs. 1 and 2, Plate LI., are the elevation and horizontal section of an 
antse finished outside door, with side lights C C, and a top^ f<^'>h <^^ traii- 
som light B. The bar A is called a transom, and this term is applied 
generally to horizontal bars extending across openings, or even across 
rooms. 

Fig. 3 is the elevation of an outside folding door. Tlie plan (fig. 4), 
shows a vestibule Y, and an interior door. The outer doors, when open, 
fold back into the pockets or recesses, ^^, in the wall. This is the present 
usual form of doors for first-class houses in this city. The fan lights are 
made semicircular, and also the head of the upper panels of the door ; 
these panels in the interior or vestibule door are of glass. 

Windows are apertures for the admission of light to the building, for 
ventihition, and for looking out. When used for the admission of light 
only, the sashes may be stationary, as they sometimes are in churches, but 



247 



PLATE LI. 



Fi- 1 






/:' .7 // .7 



AECHITECTURAL DRAWING. 



247 



for most positions tlioj are intended for all these purposes, and therefore 
the sashes are made to open, either bj sliding vertically, or laterally, or 
like doors. 

Tlie iirst is the common form of window, the sashes are generally bal- 
anced by w^eights; the second, except in a cheap form in mechanics' shops, 
are seldom used ; the third are called casements, or French windows. 

Figs. 5 and 6 represent the j)arts of the common sash window and its 
shutters, broken, so as to show the details on a large scale. Tlie general 
outside appearance of this window is familiar, and represented simply in 
the elevations, PL XLYIII. Fig. 5, Plate LL, is the elevation of the win- 
dow and shutter, in which S designates the sill of the sash frame, W the 
stone sill, with a wash to discharge the water, B is the bottom rail of the 
sash, M the meeting rails, and T the top rail, H is the head of the sash 
frame, A the architrave similar to that around doors. In the sectional 
plan, O C are the window stiles, F the pulley stile, w w the sash weights, p 
the parting strip, and D D double fold shutters. Sash windows for dwell- 
ings are almost always made with twelve lights, six in each sash. The 
height of the window must of course depend on the height of the room. 
Unless the windows begin from, or nearly from, the floor, the point a 
(fig. 5), may be fixed at a height of about 30 inches above the floor, and 
the top of the window sufiiciently below the ceiling to allow space for the 
architrave or other finish above the window, and for the cornice of the 
room, if there be any ; a little space between these adds to the effect. For 
common windows, the width of the sash is 4 inches more than that of the 
glass, and the height 6 inches more ; thus the sash of a window 3 lights 
wide and 4 lights high, of 12 x 16 glass, is 3 feet 4 inches wide, and 5 feet 
10 inches high. In plate glass windows more width is taken for the stiles 
and rails. The most usual sizes of glass are 7 x 9, 8 x 10, 9 x 12, 10 x 12, 
10 X 14, 11 X 15, 12 X 16, 12 x 18, 12 x 20, 14 x 20, but glass may be had 
of intermediate or of much larger sizes. Plate glass, either polished or 
rough, may be had of size as large as 14 x 7 feet. 

Fig. 7 represents the elevation of half of a French window, the same 
letters designate similar parts as in fig. 5. A transom bar is often framed 
between the meeting rails, and in this case the upper sash may be movable ; 
in the fig. it is fixed. An upright, called a midlion, is often introduced in 
the centre, against which the sash shuts. Fig. 8 is a section of the eleva- 
tion, ^g. 9 of the plan. 

For convenience of egi-ess and ingress, the lower sashes should not be 
less than 5 feet 6 inches high, that is, when the window opens on a stoop or 
balcony. It will be seen that in both forms of sash the bottom rail is the 



248 



AECHITECTURAL DKAWING. 



widest, and that for the same aperture the French window admits the least 
light. The chief objection to this window lies in the difficulty of keeping 
out the rain at the bottom in a driving storm. To obviate this, the small 
moulding d^ with a drip or undercut, is nailed to the bottom rail ; but the 
more effectual means is the patent weather strip, the same as used on out- 
side doors. 

The most simple exterior finish for windows in brick or stone houses, is 
a plain stone cap and sill, the height of the cap for common apertures 
being from four to five courses of brick, and the sill three courses, the 
latter always to project from one to two inches beyond the line of brick- 
work. Usually in wooden structures, and often in stone and brick, an 
architrave is formed around the window (figs. 54, and 65). For brick 
liouses the facings are made of stone. The architrave should not project 
so much as to interfere with the shutting back of the blinds. Blinds are 
commonly three-eighths of an inch narrower, and one inch longer than 
the sash. 




Fig. 54 



Fig. 55. 



Fig. 56 represents a section of the finish around the bottom of the wall 
of a room. A, is the base, consisting of a plain strip or skirting, with a 
moulding above it. B, is the surbase or chair rail ; between these, it is 
not unusual to have a panelled or plain board, called tlie dado. The 
rough plastering is usually continued to the floor, the skirting and surbase 
are then nailed on, tlic liard finish is next put on, and lastly the base 
moulding. The panels of the dado are imitated in oil or distemper ; the 
surbase is seldom used but in dining rooms or offices. 



ARCHITECTURAL DRAWING. 



240 



For the finisli of tlie angle of the wall and ceiling, the cornice adds 
often to the architectural effect. It consists of a series 
of mouldings similar to the style of finisli of the 
houses, extending around the room. Fig. 57 exhibits 
the section of quite a simple one. Tlie effect of the 
cornice is to diminish the apparent height of the 
room ; for low rooms, if adopted, it should extend in 
width on the ceiling, and but little in depth on the 
wall, and the reverse where an opposite effect is desired. 

Fire-Places. — Fire-places for wood are made with flaring jambs of the 
form shown in plan, fig, 58 ; the depth from 1 foot to 15 inches, the width 
of oj^ening in front from 2 feet 6 inches to 4 feet, according to the size of 
the room to be warmed ; height 2 feet 3 inches to 2 feet 9 inches, the 
width of back about 8 inches less than in front ; but at present fire-places 
for wood are seldom used, stoves and grates having superseded the fire- 
place. Tlie space requisite for the largest grate need not exceed 2 feet in 



Fig. 57. 



Mft 



y/ //yy/////////^'//7//////////^ ^^^ 





Fig. 58. 





Fig. 60. 

w4dth by 8 inches in depth. The requisite depth is given by the projec- 
tion of the grate, and of the mantel-piece. Fig. 60 rex^resents the eleva- 
tion of a mantel-piece of very usual proportions. Tlie length of the mantel 
is 5 feet 5 inches, the width at base 4 feet 6 inches, the height of opening 
2 feet 7 inches, and width 2 feet 9 inches. A portion of this opening is 
covered by the iron sides or architrave of the grate, and the actual open 
space would not probably exceed 18 inches in width by 2 feet in height. 
The sizes of fines are 8 x 8, 4 x 12, and 8 x 12 inches. In brick or stone 
houses the flues are formed in the thickness of the wall, but when distinct 
they have an outside shell of a half brick or 4 inches. The flues of differ- 
ent fire-places should be distinct, those from the lower stories pass up 
through the jambs of the upper fire-places, and keeping side by side with 
but half a brick betw^een them, are topped out,, sometimes in a cluster, 



250 



AECHITECTUKAL DRAWING. 



but more generally in a single line ; forming a chimney 16 inches in width 
by a length according to the number of the flues. The usual finish of the 
top of the chimney is by a stone cap and several projecting courses of 
brick, as shown in Hg. 61, or some similar form. The chimney is often 
made to add considerably to the architectural effect in its finish, as will be 
seen in Plates LXY to LXIX. 

Fig. 62. 



1 1 


III II 




[ 1 1 1 1 


1 




1 1 1 


















1 1 












1 1 






1 1 1 






1 1 






1, .11 





JFig. 61. 









/- 


^ M 


/ 

^— \ 






A 


x> 


J5 


/ 


/n\ 


\ 




Fig. 64. 



n n n 



n n n n 



Fig. 63. 



Hoof. — The general principles of the construction of roofs has been al- 
ready treated of. A front, and end elevation or gable is given, Plate XLYIII. 
But many roofs, especially of square houses, are hipped ; that is, the roof 
slopes on all sides to the eaves, as shown in elevation, fig. 63, at A and B 
the ends of the house ; at C and D there are gables. Pig. 62 represents the 
plan of the same ; lines are drawn parallel with the eaves, to show as it 
were the lap of the shingles or slates, and are generally drawn heavier or 
closer together at the ridge. Pig. 64: represents the elevation of a gambrel 
or Mansarde roof, the principle of the construction of which is given, 
p. 118. 

The eaves of roofs are finished with entablatures, consisting of various 
mouldings or members appropriate to the style of architecture of the 
structure. The gutters or eaves troughs are generally formed in the cor- 
nice (fig. (S'o)^ but sometimes they are formed into the depth of the rafter, 





rig. 05. Fig. 66. Fig. 67. 

Boinctinics on top of tlic roof (fig. 67), and sometimes by raising a parapet 



AKCHITECTUEAL DEAWING. 251 

{i^g. 65), and forming a vallei/, rather a gutter. It is desirable, however, 
that the overflow of the gutter should flow over the outer edge, and not 
back between it and the roof 

It will have been observed that for the finish of most of the parts of an 
edifice, mouldings are found necessary; so much so, that they should 
be classed among useful rather than ornamental members. These mould- 
ings are drawn either directly or indirectly from the Grecian orders of 
architecture. 

Tlie regular monldings are eight in number : Fillet or Band, Torus, 
Astragal or Bead, Ovolo, Cavetto, Cyma Eecta or Ogee, Cyma Eeversa or 
Talon, Scotia. 

To construct a Fillet. — The Jillet, a (fig. 68), is the smallest rectangular 
member employed in any composition of mouldings. When it stands on 
a flat surface, its projection is usually made eqnal to its height. It is em- 
ployed to separate members. i i 

To describe a Torus ^ or an Astragal. — ^The 1^^ Ja , 

torus and astragal are semicircles in form pro- 
jecting from vertical diameters, as in fig. 69. rig. es. rig. 

Bisect the vertical diameter a ^, on which the figure is projected ; on the 
centre <?, describe a semicircle with c a as radins. The astragal is described 
like the torus, and is distinguished from it in the same order by being 
made smaller. The torus is generally employed in the bases of columns ; 
the astragal, in both the base and capital. 

To describe an Ovolo.- — ^The ovolo is a member strong at the extremity, 
and intended to support. Tlie Roman ovolo consists of a quadrant or a 
less portion of a circle ; the Greek ovolo is elliptic. 

First, the Roman ovolo. When the projection is equal to the height. 
Draw a b for the height, and bo at right angles 
and eqnal to it, for the projection. On the 
centre b describe the quadrant c a. 

When the projection is less than the height. 
Draw a b and b c (fig. 71), as before, equal to ^'°- "^^^ ^^?- '^^• 

the height and the projection. On centres a and c, with radius a &, describe 
arcs cutting at d ; and on d with same radius describe ^ . 

the arc ado form the ovolo. "'"""'" ^^ 

Second, the Greek ovolo. Draw df from the lower 
end of the proposed curve, at the required inclination ; 
draw the vertical ^ ^y to define the projection, the 
point e being the extreme point of the curve. Draw ^^s- 'J'^- 

e h parallel to df^ and draw the vertical dhh^ such that d h is> equal to 




252 



AECHITECTUEAL DRAWING. 



2 




Fig. 73. 



Fig. 74. 



Fig. 75. 



h Ic. Divide e h and ef into the same number of equal parts ; from d 
draw straight lines to the points of division in ef^ and from h draw lines 
to meet those others snccessively. The intersections so found are points in 
the curve, which may be traced accordingly. 

To describe a Ccovetto. — ^The cavetto is described like the Roman ovolo : 

by circular arcs, as shown in 
figs. 73 and 74. Sometimes 
it is composed of two circular 
arcs united (fig. 75) ; set off 
1) e^ two-thirds of the projec- 
tion, draw the vertical 5 d 
equal to I e, and on d describe the arc h i. Join e d and produce it to j) ; 
draw in perpendicular to ed^ set oW no equal to n% and draw the 
horizontal line op meeting ep) ; on^ describe the arc io to complete the 
curve. 

To describe a Cyma recta^ or Ogee. — ^The ogee (^g. 76), is compounded 

of a concave and a convex surface. Join a 
and 5, the extremities of the curve, and bisect 
ab Sit c / on a, c, as centres, with the radius 
a c, describe arcs cutting at d; and on b, c, 
describe arcs cutting at e. On d and ^, as 
centres, describe the arcs ac, cb, composing the moulding. 

To describe a Cyma reversa^ or Talon. — ^The talon (fig. 77), like the 
ogee, is a compound curve, and is distinguished from the other by having 
the convex part uppermost. It is described in the 
same manner as the ogee. 

JSFote. — If the curve be required to be made quicker, 
a shorter radius than a c must be employed . The proj ec- 
tion of the moulding n b (fig. 76), is usually equal to the 
height a n. 

Join the extreme points a^ b (fig. 78) ; bisect 
a b at c?, and on ac, cb^ describe semicircles. Draw 
perpendiculars do^ &c., from a number of points in 
ac^cb^ meeting the circumferences ; and from the same 
points set off horizontal lines equal to the respective 
perpendiculars : o n equal io o d for example. The 
curve line b n <2, traced through the ends of the lines, 
will be the contour of the moulding. 

To describe a Scotia. — Divide the pcrpendicuhir ab 
(fig. 79), into three equal parts ; and with the first, a e, 




Fig. 77. 





AKCHITECTIJIIAL DRAWING. 253 

as radius, and on centre e^ describe the arc ctfli; on the perpendicular <?<?, 
setoff cl equal to ae^ join el and bisect it by the perpendicular c> cZ, 
meeting co at o. On centre o^ with radius o c, describe the arc c h to 
complete the figure. 



ORDERS OF ARCHITECTURE. 

Order, in architecture, is a system or assemblage of parts subject to 
certain uniform established proportions, regulated by the ofiice each part 
has to perform. An order may be said to be the genus, whereof the sj^e- 
cies are Tuscan, Doric, Ionic, Corinthian, and Composite ; and consists of 
two essential parts : a column and entablature. 

Tliese are subdivided, the first into three parts, namely : the base, the 
shaft, and the capital. The second also into three parts, namely : the 
architrave, or chief beam, C, Plate LII., which stands immediately on 
the column ; the frieze B, which lies on the architrave ; and the cornice A, 
which is the crowning or uppermost member of an order. In the subdi- 
visions certain horizontal members are used, which from the curved form 
of their edges are called mouldings, the construction of which has abeady 
been explained, and their apphcation may be seen on the Plate ; thus a is 
the ogee, 5 the corona, c the ovolo, d the cavetto, which with fillets com- 
pose the cornice, yy the fasciae. The capital of the column consists of the 
upper member or abacus ^, the ovolo moulding <?, the astragal ii^ and 
the neck h. The base consists of the torus A', and the plinth I, The 
character of an order is displayed, not only in its column, but in its general 
forms and detail, whereof the column is, as it were, the regulator ; the ex- 
pression being of strength, grace, elegance, lightness, or richness. Though 
a building be without columns, it is nevertheless said to be of an order, 
if its details be regulated according to the method prescribed for such 
order. 

In all the orders a similar unit of reference is adopted for the construc- 
tion of their various parts. Thus, the lower diameter of the column is 
taken as the proportional measure for all other parts and members, for 
which purpose it is subdivided into sixty parts, called minutes, or into two 
modules of thirty minutes each. Being proportional measures, modules 
and minutes are not fixed ones like feet and inches, but are variable as to 
the actual dimensions wliich they express — ^larger or smaller, according to 
the actual size of the diameter of the column. For instance, if the diam- 
eter be just five feet, a minute, being one-sixtieth, will be exactly one inch. 



254: AECHITECTURAL DRAWING. 

Therefore before commencing to draw an elevation of any one of the 
orders, determine the diameter of the column, and from that form a scale 
of equal parts, bj sixty divisions, and then lay off the widths and heights 
of the different members according to the proportions of the required 
order as marked in the body or on the sides of the plates. 

Plate LII., presents an illustration of the Tuscan order, considered by 
architects as a spurious or plain sort of Doric, and hardly entitled to re- 
mark as a distinct order, e^ in the frieze corresponding to the triglyph, 
illustrates still further the connection of the two orders ; but by many 
architects this member is not introduced. Fig. 1 is an elevation of cap- 
ital and entablature, ^g. 2 of the base, and fig. 3 of a detached capital. 
Our example is constructed according to the rules given by Vincent 
Scamozzi. 

Examples of two capitals are given, differing merely in the number of 
mouldings in the abacus. In fact, this introduction of simple mouldings 
is about the only variety allowable in the order. Ornament is not admit- 
ted, nor are the pillars ever fluted. 

A slightly convex curvature, or entasis, is given in execution to the 
outline of the shaft of a column, by classic architects, just sufiicient to 
counteract and correct its appearance, or fancied appearance, of curvature 
in a contrary direction {i. 6., concavely), which might else take place, and 
cause the middle of the shaft to appear thinner than it really is. 

Fig. 4 represents the form of a half column from the Pantheon at 
Pome. In ^g. 5, another example of entasis, the lower third of the shaft 
is uniformly cylindrical ; the two upper thirds are divided into seven equal 
parts. On the semicircle shown in the figure, is a chord cut off parallel 
to the diameter, the length of which is fifty-two parts, only one-half being 
shown. Divide that part, a 5, of the circumference between the diameter 
and chord into seven equal parts, and draw parallel lines from each division 
to those of the upper part of the column, which will give the diameter of 
the shaft at each division; by increasing the number of the divisions, 
more diameters for different parts of the shaft may be found. 

PL LIII. exhibits an example of the Doric order, from the temple of 
Minerva in the Island of Egina. The dimensions are given in parts of the 
diameter, as in the preceding plate, and the same capital letters denote 
corresponding parts. Fig. 1 is an elevation of the capital and the entab- 
lature. Fig. 2 of the base, and a part of the Podium. Fig. 3 shows, the 
forms of the flutes at the top of the shaft, and fig. 4 at the base. Fig. 5, 
the outline of the capital on an enlarged scale. 

The Doric order may be said to be the original of the Greek orders, 



PLATE LII. 



254 




2C=^ 



27| ---I 



rig. 4 






Fig. 5. 



^■: 



29f 



/' 



^*- 



2€;: — 



^'5:^ 



2/?f- 



ivT'/T 



PLATE LIII. 



2',i 




255 



PLATE LIY 



PPPHOpSiOPp 




2M 



AECHITECTURAL DRAWING. 255 

of wliicli there are properly but tlirec': the Doric, Ionic, and Corinthian, 
which differ in the proportion of their parts, and in some of the ornaments 
or mouldings. Of the Doric, the mutules a «, the triglyphs h J, the guttse 
or drops dd oi the entablature, the echinus/", and the annulets g g oi the 
capital, may be considered characteristic. With regard to the arrangement 
of the trigljphs, one is placed over every column, and one or more inter- 
mediately over every intercolumn (or span between two columns), at such 
a distance from each other that the metopes (?, or sjDaces between the tri- 
glyphs, are square. 

In the best Greek examj)les of the order, there is only a single triglyph 
over each intercolumn. One peculiarity of the Grecian Doric frieze is, 
that the end triglyphs, instead of being, like the others, in the same axis 
or central line as the columns beneath, are placed quite up to the edge or 
outer angle of the frieze. The mutules are thin plates or shallow blocks 
attached to the under side or soffit of the corona, over each triglyph and 
each metope, with the former of w^hich they correspond in breadth, and 
their soffits or under-surfaces are wrought into three rows of guttge or 
drops, conical or otherwise shaped, each row consisting of six guttge, or the 
same number as those beneath each triglyph. Though a few exceptions to 
the contrary exist, the shaft of the Doric column w^as generally what is 
technically called fluted. The number of channels is either sixteen or 
twenty, afterwards increased in the other orders to twenty-four ; for they 
are invariably of an even number, capable of being divided by four ; so 
that there shall always be a centre flute on each side of the column. 

PL LIY. presents an example of the Ionic order, taken from the temple 
of Minerva Polias at Athens. Pig. 1 is an elevation of capital and entab- 
lature, fig. 2 of the base, fig. 3 is a half of the plan of the column at 
the base and the top, fig. 4 an elevation of the side of the capital. In the 
proportions of its shaft, which are more slender, and the addition of a base, 
it differs from the Doric ; but the capital is the indicial mark of the order, 
by which it is immediately recognized. It is far more complex and irreg- 
ular than the other orders of capitals ; instead of showing four equal sides, 
it exhibits two fronts, with spirals or volutes parallel to the architrave, and 
narrower haluster sides (fig. 4), as they are termed, beneath the architrave. 

When a colonnade was continued in front and along the flanks of the 
building, this form of capital in the end column occasioned an offensive 
irregularity ; for while all the other columns on the flanks showed the vo- 
lutes, the end one showed the baluster side. It was necessary that the 
end column should, therefore, have two adjoining volute faces, which was 
effected by placing the volute at the angle diagonally, so as to obtain there 



256 



AECHITECTUEAL DEAWING. 



two voluted surfaces placed immediately back to back. This same diag- 
onal disposition of tlie volutes is employed for all the capitals alike, in 
Roman and Italian examples of this order. 

The capital admits of great diyersity of character and decoration — ^it 
sometimes is without necking, sometimes with ; which may either be plain 
or decorated, to suit the entire design. The capital may also be modified 
in its proportions, first as regards its general proportion to the column ; 
secondly, as regards the size of the volutes compared with the width of 
the face. In the best Greek examples, the volutes are much bolder than 
in the Boman. The spirals also of the volutes may be either single or 
manifold, and the eye or centre of the spiral may be made larger or smaller, 
fiat or convex, or curved as a rosette. 

Plate LY. represents an example of the Corinthian order, from the Arch 
of Hadrian, at Athens. This order is distinguished from the Ionic, more 
by its deep and foliaged capital than by its proportions,— the columns of 
both have bases differing but little from each other, and their shafts are 
fluted in the same manner. 

Although the order itself is the most delicate and lightest of the three, 
the capital is the largest, being considerably more than a diameter in 
height, varying in different examples from one to one and a half diame- 
ter, upon the average about a diameter and a quarter. 

The capital has two rows of leaves, eight in each row, so disposed that 
of the taller ones, composing the upper row, one comes in the middle, be- 
neath each face of the abacus, and the lower leaves alternate with the 
upper ones, coming between the stems of the latter ; so that in the first 
or lower tier of leaves there is in the middle of each face, a space between 
two leaves occupied by the stem of the central leaf above them. Over 
these two rows is a third series of eight leaves, turned so as to support the 
small volutes which, in turn, support the angles of the abacus. Besides 
these outer volutes, which are invariably turned diagonally, as in the 
^ four-faced Ionic capital, there are two other 
smaller ones, termed caulicoli, which meet each 
other beneath a flower on the face of the abacus. 
Tlie abacus itself is not, properly speaking, a 
square, although it may be said to be so in its 
general form. But instead of being straight, 
the sides of the abacus are concave in plan, be- 
ing curved outwards so as to produce a sharp 
point at each corner, which is usually cut ofl". 




Fig. 80. 



Fig. 80 represents one of the capitals of the Tower of the Winds, show- 



PLATE LY. 



256 




ARCHITECTTJEAL DRAWING. 257 

ing the earliest formation of the Corinthian capital. In this example the 
abacus is square, and the upper row of leaves of the kind called water 
leaves^ from their resemblance to those of water plants, being broad and 
flat, and merely carved upon the vase or body of the capital. 

Tlie proper Corinthian base difiers from that of the usual Ionic or Attic, 
in having two smaller scotioe, separated by two astragals : however, both 
kinds are employed indiscriminately. The shaft is fluted, in general, simi- 
larly to that of the Ionic column, but sometimes the flutes are cabled as it 
is called, that is, the channels are hollowed out for only about two-thirds 
of the up]Der part of the shaft, and the remainder cut so that each channel 
has the appearance of being partly filled up by a round staff or piece of 
rope, whence the term calling. 

The cornice is very much larger than in the other orders, — ^larger as to 
height, and consequently as to projection also. 

From this greatly increased depth of cornice, it consists of a greater 
number of mouldings beneath the corona, for that and the cymatium over 
it invariably retain their places as the crowning members of the whole 
series of mouldings. In om' illustration, square blocks or clentels are 
introduced, but often to the dentels is added a row of modillions^ imme- 
diately beneath and supporting the corona. These modillions are orna- 
mental blocks, curved in their under surface somewhat after the manner 
of the letter S turned thus, ui ; and between them and the dentels, and 
also below the latter, are ether mouldings, sometimes cut, at others left 
plain. Sometimes a plain uncut dentel land is substituted for dentels ; 
sometimes, in simpler cornices, that is omitted altogether, and plainer 
blocks are employed instead of modillions ; or else both dentels and mo- 
dillions are omitted. The dentel is not peculiar to this order, but is con- 
sidered as more properly belonging to the Ionic. 

The Comjposite Order is hardly to be considered as a distinct order, 
being but a union of the Ionic and Corinthian. Its capital consists of a 
Eoman Ionic one, super-imposed upon a Corinthian foliaged base, in which 
the leaves are without stalks, placed directly upon the body of the vase. 
In general, the entablature is Corinthian — but in a few examples it is Ionic. 

Although columns and entablatures do not of themselves, properly 
speaking, constitute an order, except they enter into the organization of a 
structure ; yet, as the Greek edifices as such, are almost entirely inappli- 
cable to purposes of the present day, we have confined our illustrations 
of the orders to the pillars and entablatures merely, remarking, that how- 
ever the Greek temples differed from each other as to the treatment of the 
order adopted, the number of columns and mere particulars of that kind, 
IT 



258 AECHrrECTUKAL DRAWING. 

they resemble each other. ]^ot only were their plans invariably parallel- 
ograms, bnt alike also as to proportion, forming a double sqnare, or being 
abont twice as much in length as in breadth. The number of the columns 
in front was invariably an even one, so that their might be a central inter- 
column ; but on the flanks of the edifice, where there was no entrance, the 
number of intercolumns was an even, and that of the columns an uneven 
one, so that a column came in the centre of these side elevations. 

As to the mode in which the front influenced the sides, by determining 
the number of columns for them, the established rule seems to have been 
to give the flanks twice as many intercolumns as there were columns at 
each end : thus the Parthenon, which is octastyle or eight columns in front, 
has sixteen intercolumns, and consequently seventeen columns, on each 
flank. In like manner, a hexastyle temple would have twelve intercol- 
umns and thirteen columns on each side. 

In the Doric order, the distances between the columns is governed 
entirely by the triglyphs of the frieze, so that there can be no medium 
between monotriglyphic and ditriglyphic inter columniation, accordingly 
as there is either one or two triglyphs over each intercolumn. But in the 
other orders there is no such restriction ; in them the intercolumns may be 
made wider or narrower, as circumstances require, from one diameter and a 
quarter to a half in width. Close spacing carries with it the expression of 
both richness and strength, whilst wide spacing produces an effect of open- 
ness and lightness, but also partakes of meagreness and weakness, owing 
to the want of sufficient apparent support for the entablature. 

Another mode of columniation and intercolumniation which has some- 
times been practised by Modern Architects, consists in coupling the 
columns and making a wide intercolumn between every pair of columns, 
so that as regards the average proportion between solids and voids, that 
disposition does not differ from what it would be were the columns 
placed singly. Supercolumniation, or the system of piling up orders, or 
different stages of columns one above another, was employed for such 
structures merely as were upon too large a scale to admit of the applica- 
tion of columns at all as their decoration, otherwise than by disposing them 
in tiers. Tliis method was afterwards adopted by the Architects of the Pal- 
ladian School. Sometimes all the three orders are employed in as many 
tiers of columns or pilasters. In other cases, the two extreme orders, — 
that is, the Doric and Corinthian, — are brought together ; in other cases 
but a single order. 

In one or two instances the Greeks employed human figures to support 
entablatures or beams ; the female figures or Caryatides, are almost uni- 



ARCHITECTURAL DRAWING. 259 

formly represented in an erect attitude, without any apparent effort to 
sustain any burden or load ; whilst the male figures, Telamones or Atlantes, 
manifestly display strength and muscular action. Besides entire figures, 
either Hermes' pillars or Termini are occasionally used as substitutes for 
columns of the usual form, when required to be only on a small, at least a 
moderate scale. The first mentioned consist of a square shaft with a bust 
or human head for its capital ; the latter of a half-length figure rising out 
of, or terminating in, a square shaft tapering downwards. Hermes' pillars 
seem to be in great favor with modern German architects, they having not 
unfrequently employed them for the decoration of windows. 

The Greek orders may be considered as the rudiments of modern 
architecture, but the forms of their buildings are almost entirely inappli- 
cable to modern purposes. The Eomans developed and matured the 
Corinthian order, and also worked out a freer and more complex and com- 
prehensive system of architecture. They introduced circular forms and 
curves not only in elevation and section, but in plan ; and while, among 
the Greeks, architecture was confined almost exclusively to external ap- 
pearance and effect, in the hands of the Eomans it was made to minister 
to internal display also. The true Koman order consists, not in any of the 
columnar ordinances, but in an arrangement of two pillars placed at a dis- 
tance from one another nearly equal to their own height, and having a 
very long entablature, which, in consequence, required to be supported in 
the centre by an arch springing from piers. This, as will be seen from 
fig. 1, Plate LYI. was, in fact, merely a screen of Grecian architecture 
placed in front of an arcade. Though not without a certain richness of 
effect, still as used by the Eomans, these two systems remain too distinctly 
dissimilar for the result to be pleasing, and their use necessitated certain 
supplemental arrangements by no means agreeable. In the first place the 
columns had to be mounted on pedestals, or otherwise an entablature pro- 
portional to their size would have been too heavy and too important for a 
thing so useless and so avowedly a mere ornament. A projecting key- 
stone was also introduced into the arch. This was unobjectionable in 
itself, but when projecting so far as to do the duty of an intermediate cap- 
ital, it overpowered the arch without being equal to the work required of 
it. The Eomans used these arcades with all the three orders, frequently 
one over the other, and tried various expedients to harmonize the construc- 
tion with the ornamentation, but without much effect. Tliey seem always 
to have felt the discordance as a blemish, and at last got rid of it, remov- 
ing the pier altogether, and substituting in its place the pillar taken down 
from its pedestal. This of course was not effected at once, but was the 



^ 



260 AEOHITECTUKAL DEAWING. 

result of many trials and expedients. One of the earliest of these is ob- 
served in the Ionic Temple of Concord, in which a concealed arch is 
thrown from the head of each pillar, bnt above the entablature, so as to 
take the whole weight of the superstructure from off the cornice between 
the pillars. When once this was done it was perceived that so deep an 
entablature was no longer required, and that it might be either wholly 
omitted, as was sometimes done, in the centre intercolumniation, or at all 
events very much attenuated. There is an old temple at Talavera, in 
Spain, which is a good example of the former expedient ; and the Church 
of the Holy Sepulchre, built by Constantino at Jerusalem, is a remarkable 
example of the latter. There, the architrave is cut off so as merely to 
form a block over each of the pillars, and the frieze and cornice only are 
carried across from one of these blocks to the other, while a bold arch is 
thrown from pillar to pillar over these, so as to take any weight from off a 
member which has at last become a mere ornamental part of the style. 

Figs. 2, 3 and 4, Plate LYI. from the Palace of Diocletian at Spalatro, 
are illustrations of the different modes of treatment of the arch and entab- 
lature. 

Perhaps the most satisfactory works of the Eomans are those which we 
consider as belonging to civil engineering rather than to architecture; 
their aqueducts and viaducts, all of which, admirably conceived and exe- 
cuted, have furnished practical examples for modern constructions, of which 
the High Bridge across Harlem Eiver may be taken as an illustration. < 

The whole history of Poman architecture is that of a style in course 
of transition, beginning with purely Pagan or Grecian, and passing into a 
style almost wholly Christian. The first form which Christian art took in 
emancipating itself from the Pagan was the Eomanesque, which afterwards 
branched off into the Byzantine and the Gothic. 

The Eomanesque and Byzantine, as far as regards the architectural 
features, are almost synoymous ; in the earlier centuries there is an orna- 
mental distinction, the Eomanesque being simply a debasement of Eoman 
art — the Byzantine being the art combined with the symbolic elements 
introduced by the new Christian religion. As commonly used, the dome 
is also considered a characteristic of the Byzantine, but this will be found 
among Eoman examples. In its widest signification, the Eomanesque is 
applied to all the earlier round arch developments, in contradistinction to 
the Gothic or later pointed arch varieties of the E'orth. In this view the 
JSTorman is included in the Eomanesque, and this distinction will be sufli- 
cient for our purpose. 

The general characteristics of the Gothic arc these : it is essentially 



PLATE LVI. 




ARCHITECTURAL DRAWING. 



261 



pointed or vertical in its tendency, in its details geometrical, in its win- 
dow tracery, in its 02:)enings, in its cluster of shafts and bases, in its suits of 
mouldings, and by the universal absence of the dome, and the substitution 
of th^ pointed for the round arch. 

The Eomanesque pillars are mostly round or square, and if square, 
generally set evenly, whilst the Gothic square pillar is set diagonally. 

Figs. 5, 6, 7, 8 and 9, Plate LYT., represent sections of Gothic pillars ; 
fig. 10 is half of one of the great western piers of the Cathedral of Bourges, 
measurino; 8 feet on each side. 

Figs. 11 and 12 are the elevations of capitals and bases and the sec- 
tions of Gothic pillars, one from Salisbury, the other from Lincoln Cathe- 
dral. Fig. 13 is a Byzantine caj^ital from the church of St. Sophia at 
Constantinople ; fig. 14 one from the palace at Gelnhausen ; fig. 15, a INTor- 
man one, from Winchester Cathedi'al, and fig. 16 a Gothic capital and base 
from Lincoln Cathedral. 

Mouldings. — " All classical architecture, and the Eomanesque which is 
legitimately descended from it, is composed of bold independent shafts, 
plain or fluted, with bold detached capitals forming arcades or colonnades 
where they are needed, and of walls whose apertures are surroimded by 
courses of parallel lines called mouldings, and have neither shafts nor 
capitals. The shaft system and moulding system are entirely separate, the 
Gothic architects confounded the two ; they clustered the shafts till they 
looked like a groiip of mouldings, they shod and capitalled the mouldings 
till they looked like a gi'oup of shafts." 

Gothic Mouldings appear in almost every conceivable position ; from 
the bases of piers and piers themselves, to the ribs of the fretted vaults 
which they sustain, scarce a member occurs which is incapable of receiv- 
ing consistent decoration by this elegant method. 

Jamh Mouldings. — In the earliest examples of Gor- 
man doorways, the jambs are mostly simply squared 
back from the walls ; recessed jambs succeeded, and are 
common in both jS'orman and Gothic architecture ; and 
when thus raised detached shafts were placed in each 
angle (fig. 81). In the later styles, the shafts were almost 
invariably attached to the structure. The angles them- 
selves were often cut or chamfered ofi", and the mouldings attached to the 
chamfer plane. The arrangement of window jambs during the successive 
periods was in close accordance with that of doorways. 

In the richer examples small shafts were introduced, wliich, rising up to 
the springing of the window, carried one or several of the arch mouldings. 




262 



ARCHITECTURAL DRAWING. 




Fig. 82. 



Yet mouldings are not nevertheless essential accessories ; many windows 
of tlie richest tracery have their mnllions and jambs composed of simple 
chamfers. 

Arch Mouldings^ even when not continuous, partook of the same 

general arrangement as those in the jambs, with greater richness of detail. 

When shafts were employed, they carried groups of mouldings more 

elaborate than those of the jambs, though still falling on the same planes. 

Capitals were either moulded, or carved with foliage, animals, &c. ; 

they always consisted of three distinct parts (fig. 82), the 

head mould A, the bell B, and the neck mould C. In 

Norman examples the head mould was almost invariably 

square ; in the later styles it is circular, or corresponding 

to the form of the pillar. 

Bases consist of the plinth and the base mouldings. 
The plinth was square in the Norman style, afterwards 
octagonal, then assuming the form of the base mouldings, 
it bent in and out with the outline of the pier. - Base mouldings were also 
extensively used round the buttresses, towers and walls of churches. 

String Courses. — ^The most usual and perhaps essential position of the 
string course is under the windows. In the Norman styles they were 
usually heavy in the outline, and displayed no particular beauty of arrange- 
ment. In the later styles they were remarkably light and elegant ; from 
restraint or horizontality, they now rose close under the siU of the window, 
and then suddenly dropping to accommodate themselves to the arch of a 
low doorway, and again rising to run immediately under the adjoining win- 
dow. In this way, the string courses frequently served the purpose of a 
drvp stone or hood moulding over doors ; occasionally the hood mould was 
continued from one window to the other. But in the later styles they were 
generally terminated in heads, flowers, or some quaint device, or simply 
returned at the springing of the arch. 

Cornices are not an essential feature in Gothic architecture. In the 
Norman and early English styles, the cornice was a sort of enlarged string 
course formed by the projection of the upper part of the wall, which was 
supported on brackets or corbels^ and hence termed the corbel table. 

The earliest moulding in Nonnan work, is a circular bead strip worked 
i out of the edges of a recessed arch, called a circular howtel 
(fig. 83). From a circular form the bowtel soon became 
pointed, and, by an easy transition, into the bowtel of one, 
two, or three fillets, all of which, with their numerous varieties, 
^j ^ performed important parts in the Gothic moulding system. 




AECHITECTUEAL DEAWING. 



263 




Fig. 84 is tlie scroll moulding, being, in fact, a simple filleted bowtel, 
with the fillet undeveloped on one side, as shown by the ^^^^M^:^^^^ 
dotted lines. If this moulding be cut in half, through 
the centre of the fillet, we have on the develoj)ed side 
the moulding now termed by carpenters the rule joint, j,.^ ^ 

which, by rounding off the corners by reverse curves, becomes the wave 
moulding. 

The ogee is also used very generally in Gothic architecture, both single 
and double, the latter formed by the junction of two ogees. 

Figs. 85 and 86 are examples of 
groupings of mouldings, fig. 85 being 
of the earlier Gothic, the filleted bow- 
tel with alternate hollows, fig. 86, of 
the perpendicular style, the hollow in 
the one case being made prominent, 
and dividing individual mouldings; 
in the latter insignificant, and as a 
separation of groups of mouldings. 

ArcJies are generally divided into the triangular-headed arch, the 
round-headed arch, and the pointed arch. Of round-headed arches there 
are four kinds, the semicircular, segmental, the stilted, and the horse-shoe. 

The stilted arch, ^g. 87, is semicircular, but the sides are carried down- 
wards in a straight line below the spring of the curve, till they rest upon 
the imposts. In the horse-shoe arch, the sides are also carried down be- 
low the centre, but follow the same curve (fig. 88). 




Fig. 85. 



Fis. 86. 




Fig. 87. Fig. 83. Fig. 89. Fig. 90. 

The pointed arch may be divided into two classes, those described from 
two centres, and those described from four. Of the first class there are 
three kinds, the equilateral, the lancet, and the obtuse. Tlie equilateral 
(fig. 89), is formed of two segments of a circle, of which the radii are 
equal to the breadth of the arch. The radii of the lancet segment are 
longer than the width of the arch, and of the obtuse, shorter. 

Of the complex arches, there is the Ogee (fig. 90), and the Tudor (fig. 
92). The Tudor arch is described from four centres, two on a level with 
the spring and two below it. 



264 



AECHITECTUEAL DRAWING. 



Of foiled arches, there are the round-headed trefoil (fig. 91), the pointed 
trefoil (fig. 93), and the square-headed trefoil arch {^g. 94). 






tf 




Fig. 91. 



Fig. 92. 



Fig. 93. 



Fig. 94 



The semi-circular arch is the Eoman Byzantine and Norman arch, the 
ogee and horse-shoe is the profile of many Turkish and Moorish domes, the 
pointed and foliated arches are Gothic. 

Domes and Vaults. — ^Both domes and vaults are found in Eoman 
works, but with the decline of Eoman power the art of vaulting was lost, 
and the churches of all Eoman Christendom remained with nothing but 
timber roofs. But among the Greek Christians, or Byzantines, it was re- 
tained, or else re-invented ; but the Greek vaulting consisted wholly of 
spherical surfaces, whilst the Eoman consisted of cylindrical ones. Figs. 
95 and 96 illustrate this distinction, fig. 95 being the elevation of a Eoman 
cyhndrical cross vault, and S-g. 96, the elevation of the roof of the church 





Fig. 95. 



Fig. 95. 



of St. Sophia at Constantinople ; and the sprouting of domes out of domes 
continues to characterize the Byzantine style, both in Greek churches and 
Turkish mosques, down to the present day. This system of vaulting has 
also been adopted in St. Paul's, London, and at St. Gene^aeve, Paris. As 
a constructive expedient the cross vault is to be preferred, as the whole 
pressure and thrust are collected in four definite resultants, applied at the 
angles only, so that it might be supported by four 
flying buttresses, no matter how slender, provided 
they were placed in the direction of these result- 
ants, and were strong enough not to be crushed by 
the pressure. 

Fig. 97 represents a compartment of the sim- 
plest Gothic vaulting, a, a, groin ribs, h, h, 5, side ribs. 
The Eomans introduced side ribs, aj^pearing 




ARCHITECTURAL DRAWING. 265 

on the inside as flat bands, and harmonizing with the similar form of 
pihasters in the walls, but they never nsed groin ribs ; the Gothic build- 
ers introduced these, and deepened the Eoman ribs. The impenetra- 
tion of vauhs, either round or pointed, produces elliptical groin lines, 
or else lines of double curvature. Yet the early Gothic architects rarely 
made their groin ribs elliptical, and never deviating from a vertical plane. 
These ribs were usually simple pointed arches of circular curvature, 
thrown diagonally across the space to be groined, and the four side arches 
were equally simple, the only care being that all the arches should have 
their vertices at the same level. The shell between, therefore, was no 
regular geometric surface. The strength depended on the ribs, and the 
shell was made quite light, often not more than six inches, while Eoman 
vaults of the same span would have been three or fom- feet. The differ- 
ence of principle being, that the Romans made their vault surfaces geo- 
metrically regular, and left the groins to take their chance ; while the early 
Gothic architects made their groins geometi'ically regular, and let the in- 
termediate surfaces take their chance. 

In the next step the groin ribs were elliptical, and when intennediate 
ribs or tiercerons were inserted, these ribs had also elliptical cuiwatures, 
but different from the gi'oins, in order that the vault of cut stone built 
upon them might have a regular cylindrical surface. In augmenting the 
number of tiercerons, and making them ramify, combinations of cii'cular 
arcs were substituted for the elliptic curves ; the surfaces of these vaults 
could not be cylindrical, but the ribs were placed very near each other, in 
order that the portion of the vault between each pair might practically be 
almost cylindrical. In the formation of the compound circular ribs three 
conditions were to be observed: — 1st, that the two arcs should have a 
common tangent at the point of meeting. 2d. Tliat the feet of all the 
ribs should have the same radius, up to the level at which they completely 
separate from each other. 8d. That from this point upwards, then- curva- 
tures should be so adjusted as to make them all meet their fellows on the 
same horizontal plane, so that all the ridges of the vaults may be on one 
level. 

The geometrical difficulty of such works led to what is called ^/rt?^ 
tracery vaulting. If similar arches spring from each side of the pillars 
(fig. 97), it is easy to perceive that the portion of vault springing from 
each pillar would have the form of an inverted concave-sided p}'ramid, 
its horizontal section at every level being square. Xow the later archi- 
tects converted this section into a circle, the four-sided 2:)yramid became a 
conoid, and all the ribs forming the conoidal surface became alike in cur- 



266 



AECHITECTUEAL DEAWING. 




vature, so that they all miglit be made simple circular arcs ; these ribs 
are continued with unaltered curvature till thej meet and form the 
ridge ; but in this case the ridges are not level, but gradually descend 

every way from the centre point (fig. 98): 

In the figure this is not fully carried out, 
for no rib is continued higher than those 
over the longer sides of the compartment, 
so that a small lozenge is still left, with a 
boss at its centre. When the span of the 
main arch h a, was large in proportion to 
that of h c, the arch h c became a very acute 
lancet arch, and scarcely admitting win- 
dows of an elegant or sufiicient size. To obviate this, the compound curve 
was again introduced, and the ribs were made less curved in their upper 
parts than in the lower. Hence the four- centred or Tudor arches. 

The four-centred arch is not necessarily flat or depressed, it can be 
made of any proportion, high or low, and always with a decided angle at 
the vertex. In general, the angular extent of the lower curve is not more 
than 65°, nor less than 45°. The radius of the upper curve varies from 
twice to more than six times the radius of the lower, but generally speak- 
ing, the greater their disproportion, the less pleasing is the sudden change 
of curvature. The projecting points of the trefoil arch are sometimes 
called cusps, often introduced for ornament merely, but serving construc- 
tively both in vaults and arches, as a load for the sides, to prevent them 
rising from the pressure on the crown. This property of arches has been 
explained, depending on the principle (p. 118), that if a polygon of rods 
be reversed, the position in which it will stand is that which it will assume 
for itself when loaded with the same weights and suspended ; and perhaps 
the equilibrium of some of the boldest vaultings was insured by experi- 
ments on systems of rods representing the ribs inverted ; and for any archi- 
tect who may wish to introduce pendants or cusps in his vaultings, this 
rule of trial will be found particularly useful. 

As vaultings, in general, were contrived to collect the whole pressure 
of each compartment into four single resultants, at the points of springing, 
leaving the walls so completely unloaded that they are required only as 
enclosures or screens, they might be entirely omitted or replaced by win- 
dows. Indeed, the real supporting walls are broken into narrow slips, 
placed at right angles to the outline of the building, and called huttresses. 
As to the e?iclosing walls, being not for support, they may be placed as 



AJRCHITECnJEAL DRAWING. 267 

the architect pleases, either at the outer or inner edge of the buttresses. 
The one method, being that adopted by the French architects, gave to 
their interiors those deep recesses, whilst the other, or English method, 
served only to produce external play of light and shade. 

The Korman buttress resembles a flat pilaster, being a mass of masonry 
with a broad face, slightly projecting from the wall. They are, generally, 
of but one stage, rising no higher than the cornice, under which they 
often, but not always, flinish with a slope. Sometimes they are carried up 
to, and terminate in, the corbel table. 

Fig. 1, Plate LYIL represents a buttress in two stages, with simple 
slopes as set-oflfs; this examj)le is somewhat narrower and projects more 
than the ISTorman buttress. 

Fig. 2 is a buttress of the Early English style, having a plain triangu- 
lar or pedimental head. The angles were sometimes chamfered off, and 
sometimes ornamented with slender shafts. In buttresses of different 
stages, the triangular head or gable is used as a finish for the intermediate 
stages. 

In the Decorated style, the outer surfaces of the buttresses are orna- 
mented with niches, as in fig. 3. In the Perpendicular style, the outer 
surface is often partially or wholly covered with panel-work tracery 
(fig. 4). 

It has been said that the buttress was a constructive expedient to re- 
sist the thrust of vaulting, but to resist the thrust of the principal vault, 
or that over the nave or central part of the church, buttresses of the re- 
quisite depth would have filled up the side aisles entirely. To obviate 
this, the system of fiyiug buttresses 'was adopted, that is, the connection 
of the interior witli the outer buttress, by an arch or system of arches, as 
shown in fig. 5. To add weight, and consequently solidity, to the outer 
piers, they were surmounted by pinnacles, rendering them thus a suffi- 
ciently steady abutment to the flying arches, which, in their turn, abutted 
the central vaults. 

An easy transition leads us from pinnacles to spires, the latter being 
but the perfect development of the former, and each requiring the assist- 
ance of the other in producing a thoroughly harmonious effect. Yet the 
spire never was a constructive expedient, or useful in any way. From the 
tower, the spire arose first as a wooden roof, and as height was one of the 
great objects to be attained, it was carried to an elevation beyond the 
mere requirements of a protection against the weather. 

The earlier towers of the Eomanesque style were constructed without 
spires. All are square in plan, and extremely similar in design. Fig. 6, 



268 AKCHITECTURAL DE AWING. 

Plate LYII., is an elevation of the tower attached to the church of Sta. 
Maria, in Cosmedin, and is one of the best and most complete examples 
of this style. Its dimensions are small, being bnt 15 feet broad and 110 
feet high ; a sufficiency of height, where buildings are not generally tall, 
to give prominence, without overpowering other objects, which renders 
these towers not only beautiful structures in themselves, but singularly 
appropriate ornaments to the buildings to which they were attached. 
These towers are the types of the later Italian campaniles, or bell-towers, 
most generally attached to some angle of churches, but sometimes de- 
tached, yet so placed that they still form a part of the church design. 
Sometimes they are but civic constructions, as belfries, or towers of de- 
fence. In design, the Gothic towers differ from the Italian campaniles. 
The campanile is square, carried up without break or offset, to two-thirds, 
at least, of its intended height ; it is generally solid to a considerable height, 
or with only such openings as serve to admit light to the staircases. Above 
this solid part one round window is introduced in each face, in the next 
story, two, in the one above this, three, then four, and lastly, five, the 
lights being separated by slight piers, so that the upper story is, virtually, 
an open loggia. 

The Gothic towers have projecting buttresses, frequent offsets, lofty 
spires, and a general pyramidal form. Fig. 7 is the front elevation of a 
simple English Gothic tower; here the plain pyramidal roof, rising at 
an equal slope on each of the four sides, is intersected by an octagonal 
spire of steep pitch. The first spires were simple quadrangular pyramids, 
afterwards the angles were cut off, and they became octagonal, and this is 
the general Gothic form of spire. Often instead of intersecting the square 
roof as in the figure, the octagonal spire rests upon a square base, and the 
angles of the tower are carried up by pinnacles, or the sides by battlements, 
or by both, as in ^g. 8, to soften the transition between the perpendicular 
and sloping part. 

In general the spires of English churches are more lofty than those on 
the Continent. The angle at the apex in the former being about 10° and in 
the latter, about 15°. The apex angle of the spires of Chichester and Lich- 
field, are from 12° to 13°, or a mean between the two proportions, and acr 
cording to Ferguson, more pleasing than either; although having more lofty 
spires, yet the English construction is much more massive in appearance, 
than the Continental ; the apertures are less numerous, and the surfaces are 
less cut up, and covered with ornaments. The spires of Friberg Church 
and many others on the Continent are made open work, a precedent fol- 
lowed sometimes in this country, but not in the same material — Avood 



PLATE LVIL 



26S 




AKCHITECTURAL DRAWING. 2G9 

rather tlian stone. In cast iron, the same effect would be obtained at a less 
cost, and equally durable with stone. 

Sometimes the central spires of the tower were omitted ; each of the 
pinnacles at the angles being converted as it were into spires. Sometimes 
the tower is abruptly ended by mere battlements around its sides. In the 
poorer churches, a bell-cot w^as made to serve the purpose of a bell tower ; 
this was formed by carrying up the gable wall, as in lig. 9, and making 
apertures for the recej^tion of the bell. When the wall was not of the re- 
quisite thickness, the cot was either supported by buttresses from beneath, 
or the corbels were projected from each side of the wall. 

Fig. 10 represents the upper portion of the tower of Ivan Yeliki at 
Moscow. The Kussian towers are generally constructed independent of 
their churches, and are intended for the reception of their massive bells. 

Windoios. — Before the use of painted glass, very small apertures suf- 
ficed for the introduction of the required quantity of light into a church ; 
as a consequence the windows of the Eomanesque churches were gener- 
ally small, and devoid of tracery. Again, as the Byzantine architects 
adorned their walls with paintings, they could not make use of stained 
glass ; neither in their climate, did they require large apertures ; they fol- 
lowed in general form the Eomanesque window, apertures with circular 
heads, either single or in groups (fig. 1, Plate LYIII. or fig. 6, Plate LYII). 
The Gorman windows were also' small, each consisting of a single light, 
semicircular in the head, and placed as high as possible above the gromid ; 
at first splayed on the inside only, afterwards the windows began to be 
recessed with mouldings and jamb shafts in the angles, as in fig. 2. 

The Lancet in general use in the early Gothic period was of the sim- 
plest arrangement : in these windows the glass was brought within three 
or four inches of the outside of the wall, and the openings were widely 
splayed in the interior. The proportions of these windows vary consider- 
ably ; in some the height being but five times the width, in others as much 
as eleven ; eight or nine times may be taken as the average. Lancet win- 
dows occur singly : in groups of two, three, five and seven, rarely of four 
and six. The triplet, fig. 3, is the most beautiful arrangement of lancet 
windows. It was customary to mark with greater importance the central 
light, by giving it additional height, and in most cases increased width 
also. In some examples the windows of a lancet triplet are placed within 
one dripstone forming a single arch, thus bearing a strong resemblance to 
a single three-light window. The first approximation to tracery appears to 
have been the piercing of the space over a double lancet window com- 
prised within a single dripstone ; in place of the customary simjDle 



270 AECHITEGTUEAL DRAWING. 

arch head, in some examples of lancet windows, the head of the light-is 
foiled. 

From the combination and foiling, or cusping, of distinct lancets, a 
single window divided by muUions and tracery derives its origin. 

A traceried window may be justly regarded as a distinctive character- 
istic of Gothic architecture. With the decided establishment of the prin- 
ciple of window tracery, it became a recognized constructive arrangement 
to recess the muUions from the face of the wall in which the window arch 
was pierced, and the fine effect thus produced was, as the art advanced, 
speedily enhanced by the introduction of distinct orders of mullions, and 
by recessing certain portions of the tracery from the face of the primary 
mullions and their corresponding tracery bars. The tracery bars are those 
portions of the masonry of the window head which mark out the principal 
figures of the design; from these the minor and more strictly decorative 
parts of the stone work may be distinguished under the title of Form 
pieces. 

Decorated window tracery has been generally divided into two chief 
varieties, Greometrical and Flowing; the former consisting of geometrical 
figures, as circles, trefoils, quatrefoils, curvilinear triangles, lozenges, &c. 
&c. ; while in flowing tracery, these figures, though still existing, are 
gracefully blended together in one design. In its most perfect state, geo- 
metrical tracery invariably exhibits some large figure of a distinct and de- 
cided character, which occupies the entire upper part of the window-head. 

Fig. 4 represents a quatrefoil window, fig. 5, a pointed trefoil in out- 
line ; with the centres of the different circles indicated, and such lines as 
may be necessary to explain the way in which they are described. Tliese 
forms and modifications of them, will be found of general application in 
traceried windows. Fig. 6 represents two forms of circular windows, or 
roses tournantes. 

Fig. 7 represents an example of the earlier decorated tracery window- 
head, consisting of two foiled lancets, with a pointed quatrefoil in the 
spandrel between them. One half of the windows in this, as in some of 
the following figures, is drawn in skeleton to explain their construction. 

Fig. 8 is another example of Decorated tracery. 

Fig. 9 is an example of the English leaf tracery ; fig. 10 of the French 
flamboyant. The difference between the two stylos is, that while the 
upper ends of tlie English loops or leaves are round, or simply pointed, 
the upper ends of the latter terminate like their lower ones, in angles of 
contact, giving a flame-like form to the tracery bars and form pieces. 

In England the rerpendicular style succeeded the Decorated ; the mul- 



PLATE LYIII. 



270 




271 



rLATE LIX„ 







ARCHITECTUEAL DRAWING. 271 

lions instead of diverging in flowing or curvilinear lines, are carried up 
straight tlirongli the head of the windows ; smaller niullions spring from 
the head of the principal lights, and thus the upper portion of the window 
is filled with panel-like compartments. The principal as well as the sub- 
ordinate lights are foliated in their heads, and large windows are often di- 
vided horizontally by transoms. Tlie forms of the window arches vary 
from simple pointed, to the complex four-centred, more or less depressed. 

Fig. 11 is an example of Perpendicular windows. 

Fig. 12 is a square-headed window, such as were usual in the clere 
stories of Perpendicular architecture. 

Figs. 13 and 14 are quadrants of circular windows, used more especial- 
ly in France, for the adornment of the west ends and transepts of the ca- 
thedrals. 

Besides the tracery characteristic of Gothic architecture, there is a 
tracery peculiar to the Saracenic and Moorish style, of which fig. 15 may 
be taken as an example — it being a window of one of the earliest mosques. 
The general form of the window and door-heads of this style is that of the 
horse-shoe, either circular or pointed. 

Doorioays. — Plate LIX. ^g. 1, is the elevation of a circular-headed 
doorway, which may be considered the type of many entrances both in 
Romanesque, Gothic, and later styles. It consists of two or more recessed 
arches, with shafts or mouldings in the jambs. Li the earlier styles the 
arches were circular, in the later Gothic, generally pointed, but some- 
times circular ; in the earlier, the angles in which the shafts are placed are 
rectangular ; in the later, the shaft is often moulded on a chamfer plane, 
that is, a plane inclined to the face of the wall, generally at an angle of 
45° ; often the chamfer and rectangular planes are used in connection. 

Fig. 2 is a simple head of a depressed four-centred or Tudor-arched 
doorway, with a hood moulding. Fig. 3 represents the incorporation of 
a window and doorway. Sometimes the doorway pierces a buttress ; in 
that case, the buttress expands on either side forming a sort of porch. Tlie 
Gothic architects placed doors where they were necessary, and made them 
subservient to the beauty of the design. 

Fig. 4 is an example of a gabled doorway with crockets and finials. 
Fig. 5, of a Perpendicular doorway, with a label or hood moulding above, 
and ornamented spandrels. 

Fig. 6 is an example of a Byzantine, and fig. 7 of a Saracenic doorway. 

The Renaissance style succeeded the Gothic, being, originally, but the 
revival or a fair rendering of the classical orders of architecture, with or- 
naments from the Byzantine and Saracenic styles. 



272 AUCHITECTUEAJL DRAWING. 

It was in Italy that this revival took place, and Garbett divides this 
style into three Italian schools, the Florentine, Yenetian, and Eoman, ex- 
hibiting a certain analogy to the three orders of ancient architecture. 
The Florentine, corresponding to the Doric, admits of little apparent 
ornament, but any degree of real richness, preserving in its principal 
forms severe contrast; powerful masses self-poised without corbelling, 
without arching ; hreadth of every thing, of light, of shade, of ornament, 
of plain wall ; dejpth of recess in the openings, of perspective in the whole 
mass, of projection in the cornice. To these add a sort of utilitarianism, 
or absence of features useless to convenience or stability, an absence of 
sacrifice of material^ admitting of great plainness, of very fiorid enrich- 
ment. On the whole, the Florentine may be called the plain, common- 
sense school. 

Yery different in principle was the Yenetian school, which, like its 
prototype, the Corinthian, superseded its sober rivals. Its aim was splen- 
dor, variety, show, and ornament ; not so much real as effective ornament. 
Thus, it rarely contains so much carving or minute enrichment as the Flo- 
rentine admits ; but it has larger ornaments, constructed (or built) orna- 
ments, great features useless except for ornament, as inaccessible porticoes, 
detached columns, and architraves supporting no ceiling, towers built only 
for breaking an outline. Its decoration is spread equally over the whole 
work. Rectangular severity gives place to curved elegance, in arches, 
domes, circular and oblique-angled plans, true grandeur to effect, intellec- 
tual sense of fitness to eumorphic beauty. 

The Roman school, holding the same place as the Ionic, is intermediate 
in every respect between the two other schools. It is better adapted to 
churches than to any other class of buildings. This fitness arises from the 
grand, simple, p,nd unitory effect of one tall order, generally commencing 
at or near the ground, and including, or rather obliterating, the distinction 
of two or three stories, making a high building appear a single story. 

To describe these schools technically, the Florentine is mostly astylar, 
the style of finistration and rustic cpoins ; the Roman, the style of pilas- 
ters ; and the Yenetian, that of columns. In calling the Florentine asty- 
lar, a total absence of external orders is not implied, but their absence as 
main features, or on a considerable scale. Their chief application is to 
windows and doors, and the greatest orders never include so much as the 
height of a single story. In the Roman school, the great scale of the 
principal order renders it chiefly an order of pilasters. The outer pilasters 
of the great order were often filled in Avith smaller and columnar ones, in 
two tiers, while a still smaller set decorated the openings. As the Yene- 



AECHITECTTJEAL DRAWING. 



273 



tians did not use such large orders, they easily made them more columnar, 
and introduced hanging entablatures. In this school there is (except in 
churches) no principal story or order, if there be more than one, all are 
nearly equal, or equally important. 

General plan and outline, in the Florentine, is of the utmost simplicity, 
rendering it fitter for town than country buildings ; in the Eoman, slightly 
more varied ; in the Venetian, whenever the site will admit, broken, 
complex, and picturesque. 

We have thus briefly treated of the distinguishing features, according 
to Garbett, of the three modern schools ; there are many other distinctive 
styles and names, but they may mostly be included under the one or the 
other of these schools, their claim for a distinctive name resting rather on 
the peculiar style of ornament or tracery used, than any great distinctive 
architectural feature. 

Ornament. — Architectural ornament is of two kinds, constructive and 
decorative. By the former is meant all those contrivances, such as capi- 
tals, brackets, vaulting shafts, and the like, which serve to explain or give 
expression to the construction ; by the latter, such as mouldings, frets, 
foliage, ttc, which give grace and life, either to this actual constructive 
forms, or to the constructive decoration. It is to the latter class that we 
wish to call attention ; mouldiugs of the different styles have been already 
treated of; we therefore propose to give now what are even more merely 
decorations of a style. 

First, as to Grecian orders. By reference to Plate LIII. we see that 
the Doric has the triglyph mutules and guttse. By reference to Plate 
LIY., the Ionic, we find various mouldings of the cornice fiieze, abacus, 
and neck of the column enriched. The princi]3al ornament of the neck 
of the column is the anthemion, commonly known, in its most simple 
form, as the honeysuckle or palmetto ; in the anthemion as represented in 
the figure, the palmetto alternates with the lily or some analogous foi-m. 
Tlie ornament of the abacus is the Q^g and dart, shown on a large scale, 
"^g. 9, Plate LX., where may be found also the 
ornament of the frieze and cornice, fig. 7. Fig. 99, 
the fret, and fig. 100, the guilloche, are also com- 
mon Greek ornaments, used to adorn the soffits 
of beams, and ceilings. The acanthus is the dis- 
tinctive ornament of the Corinthian, of which a 
leaf is represented on a large scale in front and 



a 






Fisr. 



side view, figs. 1, 2, 3, Plate LX. 



Tliese figure: 




illustrate, also, the 



way m 

18 



which ornaments of 



Fig. 100. 



274 AECHITECTURAL DRAWING. 

irregular figure are copied bj the drauglitsman. Thus, suppose it were 
required to draw 'Q.g. 2, and in a reversed position ; circumscribe around 
tlie given figure, a parallelogram ; divide this parallelogram into any 
number of equal squares, and in the position required for the copy, as 
fig. 3 for instance, construct a similar parallelogram. For convenience 
of reference, we have marked the vertical divisions of the squares by let- 
ters, a, h, Cy d, e,f, g, and the horizontal divisions by figures, 1, 2, 3, 4 ; this 
is done with both parallelograms. But it must be remarked, that if the 
copy is to be a reverse of the original, the figures marking the horizontal 
divisions of the copy must be the reverse of the original, as may be seen 
in figs. 2 and 3. Now mark on the difierent verticaland horizontal lines 
of the corresponding squares, the relative positions of the parts of the 
leaf, and through these points thus established, construct the leaf required. 
A similar method may be used in constructing a copy to an enlarged 
or to a reduced size of the original, by enlarging or reducing the compar- 
ative sizes of the squares of the parallelogram of the copy on a scale pro- 
portioned to the enlargement or reduction required. In general, the inter- 
sections of the portions of the leaf, or other figure, with the vertical or 
liorizontal lines, are measured and transferred by the eye ; the larger the 
number of squares, therefore, the greater probability of the copy coinciding 
with the original. Figs. 4, 5, and 6, are the side elevation, front elevation, 
and section of a Greek bracket, the principal ornaments of which are taken 
from the anthemion and acanthus. 

Fig. 1, Plate LXI., is an elevation of a portion of an enriched cornice 
from the temple of Jupiter Stator at Rome, of the Corinthian order of ar- 
chitecture. Fig. 2, is the under side of the modillion. 

Tlie chief characteristic of Roman ornament, is its uniform magnifi- 
cence. As a style it is not original, but rather an enlargement or enrich- 
ment of the Greek. There is, further, this distinction between the two 
styles, that the most rarely used elements among the Greeks are the most 
characteristic of the Roman decorations, the scroll and the acanthus. In- 
deed, every form which will admit of it, is habitually enriched with acan- 
thus clothing or foliations. The acanthus of the Greeks is the narrow 
prickly acanthus ; that of the Roman, the soft acantlius. For capitals the 
Roman acanthus is commonly composed of conventional clusters of olive 
leaves. The Greek scroll is seldom elaborated, but the Roman is seldom 
without acanthus foliations. Fig. 3, represents a Roman acanthus scroll. 

The free introduction of monsters and animals is likewise a character- 
istic of Greek and Roman ornament, as the sphinx, the triton, the grifiin, 
and others ; they occur however more abundantly in the Roman. 



PLATE LX 




PLATE I.XI. 



274 



Fig. 2. 



^ 







Fi-. 1. 



V^IVU;!AUJIU4VU4UJ41UJI\IPJ 



^/. 



irmirifiraffriii 



^^^ 



^mMk 



Fis. 3. 









Fig. 3. 






m 




P 


1^ 


1 


i^jS 




^^S 




^ 



AECHITECTUEAL DRAWING. 275 

As tlie Cliristian art succeeded the Pagan, symbols became the founda- 
tion of decorations in the Byzantine and Eomancsque. Tlie early sym- 
bols were the monogram of Chi'ist, the lily, the cross, the serpent, the fish, 
the aureole, or vesica piscis, and the circle or nimbus, the glory of 
the head, as the vesica is of the whole body. Tliese are very important 
elements in Christian decoration, especially the nimbus, which is the ele- 
ment of the trefoil and quatrefoil ; the first having reference to the Trinity, 
the second to the four Evangelists, as the testimony of Christ, and to the 
Cross, at the extremities of which we often find four circles, besides the 
circle in the centre, which signifies the Lord. Occasionally the symbolic 
images of the Evangelists, the angel, the lion, the ox, and the eagle, ai-e 
represented within these circles. 

Tlie hand in the attitude of benediction, and the lily (the fleur-de-lis), 
the emblem of the virgin and purity, are common in Christian decoration. 
This last symbol was eventually elaborated into the most characteristic 
foliage of Byzantine and Eomanesque art. Conspicuous in their foliage 
also is a peculiar formed leaf, somewhat resembling the leaf of the ordinary 
thistle. The serpent figures largely in Byzantine art as the instrument of 
the fall, and one type of the redemption. 

As paganism disappeared, their ornaments, under certain symbolic 
modifications, were admitted into Christian decorations. Thus the folia- 
tions of the scroll were terminated by lilies, or by leaves of three, four and 
five blades, the number of blades being significant ; and in a similar way, 
the anthemion and every other ancient ornament. In the Byzantine the 
symbolism is seldom or ever absent, however much it may be modified or 
disguised. An important featm-e, always to be observed in the Byzantine, 
is that all their imitations of natural forms were invariably conventional ; 
it is the same even with animals and the human figure, every saint had his 
prescribed colors, proportions and symbols. 

Tlie Saracenic was the period of gorgeous diapers, for their habit of 
decorating the entire surfaces of their apartments was highly favorable to 
the development of this class of design. The Alhambra displays almost 
endless specimens, and all are in relief and enriched with gold and color, 
chiefly blue and red. The religious cycles and symbolic figures of the By- 
zantine are excluded. Mere curves and angles or interlacings were now 
to bear the chief burden of a design, but distinguished by a variety of co- 
lor. Tlie curves however very naturally fell into standard forms and floral 
shapes, and the lines and angles were soon developed into a very charac- 
teristic species of tracery, or interlaid strap work, very agreeably diversified 
by the ornamental introduction of the inscriptions, which last custom of 



276 AKCHITECTUEAL DEAWING. 

elaborating inscriptions with their designs was peculiarly Saracenic. Al- 
though flowers were not palpably admitted, yet the great mass of the 
minor details of Saracenic designs are composed of flower forms disguised, 
the very inscriptions are sometimes thus grouped as flowers ; still no ac- 
tual flower ever occurs, as the exclusion of all natural images is funda- 
mental to the style in its purity. 

Fig. 3, is a specimen of Alhambra diaper. 

All the symbolic elements of the Byzantine are continued in the Gothic. 
Ornamentally, the Gothic is the geometrical and pointed element elaborated 
to the utmost ; its only peculiarities are its combinations of details ; at first 
the conventional and geometrical prevailing, and afterwards these com- 
bined with the elaboration of natural objects in its decoration. The By- 
zantines never did this, their ornaments are purely conventional; while 
in the finest gothic specimens, not only the traditional conventional orna- 
ments, but also elaborate imitations of natural plants and flowers are found. 
The most striking feature of all Gothic work is the wonderful elaboration 
of its geometric tracery ; vesicas, trefoils, quatrefoils, cinquefoils, and an 
infinity of geometric varieties besides. The tracery is so paramount a 
characteristic, that the three English varieties, the early English, the deco- 
rated, and the perpendicular, and the French flamboyant, are distinguished 
almost exclusively by this feature. See Plate LYIII. 

Under the head of Gothic, the E'orman is often included, but it is rather 
a transition style between the Romanesque or Byzantine, and the Gothic. 
The ornamental mouldings used in the decorative details of this style are 
numerous, among which the more common is the chevron or zig-zag, (fig. 
1, plate LXII.,) simple as the indented, or duplicated, triplicated or quad- 
rupled ; the billet, the prismatic billet, the square billet, and the alternate 
billet (fig. 2) ; the star -Gig, 3, the fir cone ; the cable, fig. 4 ; the embat- 
tled, fig. 5 ; the nail head, fig. 6. In the early English style we find the 
dog-tooth, fig 7 ; a kind of j)yramid-shaped flower leaves ; the ball flower, 
fig. 8, and the serpentine vine scroll, are the most characteristic ornamen- 
tal mouldings of the decorated style. The mouldings of the perpendicu- 
lar are not peculiar ; they are less enriched than the preceding styles, and 
the same panelling which is found in the windows is spread over every 
surface of the building. 

In tlie early English we have the first development of geometrical tracery, 
flying buttresses, crocheted pinnacles, columns clustered, and an extensive 
application of foliage with the trefoil leaf, as the most characteristic orna- 
ment ; sometimes formed as a clover leaf, at other times very irregularly 
formed. 



m 



PLATK LXli. 



Jiu'. 1. 







<^^^ 




■y^ 



Fijr. 3. 



Fig. 5, 



Fi- G. 




ARCHITECTURAL DRAWING. 277 

The early English is characterized, besides its tracery, by the ogee and 
the pinnacled canopied recesses of its buttresses and other parts producing 
a prominence of diagonal lines. There is also more co])ying of nature in 
its ornamental details. 

In the Perpendicular, the new features are the horizontal line and the 
panellings, and the substitution of perpendicular for flowing tracery. 

The crocket, in its earliest form, was the simple arrow head of the 
Episcopal, pastoral staff; subsequently finished with a trefoil, and after- 
wards still further enriched. Figs. 9 and 10 are early English crockets ; 
fig. 11 a decorated one. Fig. 12 is a finial of the same style ; both finials 
and crockets in detail display a variety of forms ; some resembling the 
botanical productions of one class, some of another. 

The Parapets of the early English style are often a simple horizontal 
course, supported by a corbel table, sometimes relieved by a series of sunk 
blank trefoil-headed panels ; sometimes a low embattled parapet crowns 
the wall. In the decorated style, the horizontal parapet is sometimes 
pierced with trefoils, sometimes with wavy flowing tracery (fig. 13). G-ro- 
tesque spouts or gargoyles discharge the water from the gutters. The 
parapets of the perpendicular style are frequently embattled (14), covered 
with sunk or pierced panelling, and ornamented with quatrefoil, or small 
trefoil-headed arches ; sometimes not embattled but covered with sunk or 
pierced qnatrefoils in circles, or with trefoils in triangular spaces as in 
^g. 15. 

Amongst the varieties of ornamental work, the mode of covering small 
plain surfaces with diapering (fig. 16), was sometimes nsed ; the design 
being in exact accordance with the architectural features and details of 
the style. The rose, fig. IT, the badge of the houses of York and Lancas- 
ter, is often met with in the perpendicular st^de ; and tendrils, leaves and 
fruit of the vine, are carved in great profusion in the hollows of rich cor- 
nice mouldings, especially on screen work in the interior of a church. 
Fig. 18, in its original type, a Byzantine ornament, an alternate lily and 
cross, is a common finish to the cornice of rich screen work in the latest 
Gothic, and is known mider the name of the Tudor flower. 

Figs. 19, 20, 21, are examples of ornamental crosses nsed as finials, 
either for spires or pinnacles. 

The Ornaments of the Henaissance. — ^Tlie term Penaissance is used in a 
double sense ; in a general sense implying the revival of art, and specially, 
signifying a peculiar style of ornament. It is also sometimes, in a veiy 
confined sense, applied in reference to ornament to the style of Benvenuto 
Cellini ; or, as it is sometimes designated, the Henry 11. (of France) style. 



278 AECHITECTUEAL DRAWING. 

The mixture of various elements is one of the essentials of this style. 
These elements are the classical ornaments ; unnatural and natural flowers 
and foliage, the former often of a pure Saracenic character; man and ani- 
mals, natural and grotesque ; cartouches, or pierced and scrolled shields, in 
great prominence ; tracery independent, and developed from the scrolls of 
the cartouches ; and jewel forms. Fig. 1, and 3, Plate LXIII. 

The Elizabethan is a partial elaboration of the same style, the only 
difference being that what we now term the Elizabethan exhibits a very 
striking preponderance of strap and shield work, but the earlier and pure 
Elizabethan is much nearer allied to the continental styles of the time; 
classical ornaments but rude in detail, occasional scroll and arabesque 
work, and strap work, holding a much more prominent place than the 
pierced or scrolled shields. Fig. 2 is an example of the style from the old 
guard chamber, Westminster. 

Of the earliest and transition styles of Henaissance ornament, are the 
Tricento and the Quatrecento ; the great features of the first are its intricate 
tracery and delicate scroll work of conventional foliage, the style being 
but a slight remove from the Byzantine and Saracenic. Of the second 
are, in addition, elaborate natural imitations of fruit, flowers, birds or 
animals (fig. 4), all disposed simply with a view to the ornamental ; also 
occasional cartouches, or scrolled shield work. 

In all these styles, the evidence of their Byzantine or Saracenic origin 
is constantly preserved, in the tracery, in the scroll work and foliage, and 
in the rendering of classical ornaments. The Benaissance is, therefore, 
something more approximative to a combination of previous styles than a 
revival of any in particular. Yet it is a style that w^as developed solely 
on esthetic principles, from a love of the forms and harmonies themselves, 
as varieties of effect and arrangements of beauty, not because they had 
any particular signification, or from any superstitious attachment to them 
as heirlooms. 

Fig. 5 is an example of ornament in the Cinquecento style. Tlie ara- 
besque scroll work is the most prominent feature of the Cinquecento, and 
with this in its elements, it combines every other feature of classical art, 
with the unlimited choice of natural and conventional imitations from 
the entire animal and vegetable kingdom, both arbitrarily disposed and 
coml)ined. Absolute works of art, such as vases and implements, and in- 
struments of all kinds, are prominent elements of the Cinquecento ara- 
l)esque, but cartouclies and strap work wholly disappear from the best 
examples. Anotlicr cliief feature of the Cinquecento is the admirable 
play of color in its arabesques and scrolls, and it is worthy of note that 



PLATE LXIir. 



278 



^^ Ti},'. 1. 




AKCHrrECTUKAL DRAWING. 279 

the thi-ee secondary colors, orange, green, and puri^le, perform the chief 
parts in all the colored decorations. 

Fig. 6 is an example of the Louis Quatorze style of ornament. The 
great medium of this style was gilt stucco work, and this absence of color 
seems to have led to its most striking characteristic, infinite play of light, 
of shade ; color, or mere beauty of form in detail, having no part in it 
whatever. Flat surfaces are not admitted ; all are concave or convex : this 
constant varying of the surface gives every point of view its high lights 
and brilliant contrasts. 

The Louis Quinze style differs from that of Louis Quatorze chiefly in 
its absence of symmetry ; in many of its examples it is an almost random 
dispersion of the scroll and shell, mixed only with that peculiar crimping 
of shell work, the coquillage. 

The ornaments of which we have thus given examples are, in general, 
applied to interior decorations, to friezes, pilasters, panels, architraves, the 
faces and soffits of arches, ceilings, &c., to furniture and to art manufac- 
tures in general. For exteriors these ornaments are sparingly applied ; 
shield and scroll work, of the later Elizabethan or Eenaissance style, is 
sometimes used, but very seldom tracery. 

Of common exterior ornament, the baluster is peculiarly modern, with 
all the refinement of a classic model. Balustrades are sometimes of real 
use in building, and at other times merely ornamental. Such as are in- 
tended for use, as when they are employed on steps or staii's, before win- 
dows, or to enclose terraces or other elevated places of resort, must always 
be nearly of the same height, from three to three and a half feet, so that 
a person of ordinary height may, with ease, lean over them without the 
danger of falling. But those that are principally designed for ornament, 
as when they finish a building ; or even for use and ornament, as when 
they form the railing over a large bridge, should be proj^ortioned to the 
architecture they accompany, and their height ought never to exceed four- 
fifths of the entablature on which they are placed ; nor should it be less 
than two-thirds, without counting the plinth, the height of which must be 
sufficient to leave the whole balustrade exposed to view. 

Figs. 101, 102, 103, 104, 105, and 106, 
represent various figures of balusters and of va- 
rious proportions, suited to the various orders 
they may serve to finish. Tlie double-bellied 
balusters (figs. 101 and 102) are the lightest, 
and, therefore the best adapted to windows ^^=-'''- ^'-''- ^'-^''- ^^-'^ 
or other compositions of which the j^arts are small and the profiles 




^ 



280 



AKCHITECTUEAL DRAWING. 



delicate. The base and rail may be of the same profile but not so large 
as for single-bellied ones. 

In balustrades, the distance between two balus- 
ters should not exceed the half of the diameter of 
the thickest part of the baluster, nor less than one- 
third of it. The pedestals, if possible, should be at 
intervals of about nine balusters, but as the pedes- 
tals must be placed over the centre of the piers, the 
intervals must frequently contain more balusters. 
Fig. 105 shows the arrangement of a baluster 
with inclined rails and bases. 

"When used in interiors, either for decoration or use, the forms of the 
baluster are much varied and enriched ; this is especially observable in 
constructions in iron. 




Fig. 105. 



Fig. 106. 



ELEVATIONS OF HOUSES, 



Having thus given a brief abstract of the characteristics of various 
prominent styles of architecture, we continue our article on houses by 
giving elevations, either suited to plans already exhibited, or to other 
plans which will be found, on the same plate as the elevations. It must be 
perceived that, in general, in modern constructions, pure ancient art is 
seldom exhibited, nor would it, in domestic architecture, be found suitable, 
the requirements and appliances being very different, and he may be 
called an architect, who, conversant with ancient and modern practice, 
can adapt them in unity and harmony to modern necessities. 

PL LXIY. represents the front elevation of a basement house with the 
general characteristics of the Florentine style, uniting richness and gran- 
deur of effect, admirably suited to the locality and purpose for which it 
is designed, a first-class house, or even what might be termed a palatial 
residence. This building has been constructed in Fifth Avenue, J^ew 
York, after designs of T. Thomas & Son, architects ; the specifications of 
which will be found in a subsequent chapter. 

Miglish hasement houses are generally constructed with a rusticated 
basement as in the preceding example, (PL LXIY.) with a balustrade 
marking the distinction between it and the principal story. The entrance 
is generally raised not to exceed three steps, and seldom with a j)rqjecting 
porch ; the intention being to make the basement subordinate to the prin- 
cipal story, the usual finish of the door-head is similar to that of the win- 
dow. In general English basement houses are intended for narrow lots ; 



PLATE LXIY. 



2S0 




2S1 



PLATE LXYL 




WOOD 
l-l O U S E 
P XI2 



MILK 

Riyi 
8>(g 




281 



PLATE LXYIL 





AECHITECTUEAL DRAWING. 281 

sliowing in front but two windows to the stories above the basement, and 
one basement window. CircuLar heads are almost invariably used for the 
basement openings, and for the windows above either square or slightly 
arched lintels. Sometimes a species of Eomanesque window of clustered 
openings is adopted. 

"When the architect is not controlled by the form or size of the lot, 
much picturesqueness may be given by varieties of form and irregularities 
of outline in the construction of edifices. 

Plate LXY. is an elevation of a house from Holly's " Country Seats," 
after a style of architecture usually designated here as the French, from 
the form of the Mansard roof and its dormer windows, rather than any dis- 
tinctive features in the main elevation or its ornaments. This style of roof 
is very effective, and has become very popular ; it is well adapted both for 
city and country residences. 

Plates LXYI., LXYII., LXYIII., and LXIX. are elevations and plans of 
countr}^ residences from " Downing's Country Houses," drawn in perspec- 
tive, the principles of which will be given in a subsequent chaj)ter. They 
may be taken as beautiful illustrations of modern constructions. 

Plate LXYI. are the plans and elevation of a Farm House in the Eng- 
lish Rural Style. 

Plate LXYII. is an elevation and plan of a plain timber cottage villa, 
after designs of Gervase Wheeler, Architect, of Philadel23hia. "The 
construction, though simple, is somewhat peculiar. It is framed in such a 
manner that on the exterior the construction shows. At the corners are 
heavy posts, roughly dressed and chamfered, and into them are morticed 
horizontal ties, immediately under the springing of the roof ; these, with 
the posts and the studs and the framing of the roof showing externally. 
Internally are nailed horizontal braces at equal distances apart; sto23ping 
on the posts and studs of the frame, and across these the furring and lath- 
ing cross diagonally in different directions. On these horizontal braces, 
the sheathing composed of plank placed in a j)erpendicular position is sup- 
ported and retained in its place by battens, two and a half inches thick, 
and made wdth a broad shoulder. Tliese battens are pinned to the hori- 
zontal braces, confining the planks, but leaving spaces for shrinking and 
swelling, thus preventing the necessity of a single nail being driven 
through the planks. A representation is given (fig. 107) of the batten B, 
and the mode of framing. 

Fig. 108 represents the usual form of vertical boarding, which is less 
expensive than the first illustration, and, in general, will be found suffi- 
ciently secured for the class of buildings to which it is applied. 



282 ARCHITECTUKAL DRAWING. 

Plate LX YIIl. is a villa in what Mr. Downing designates as the Knral 




Fi2. 107, 



Fig. 108. 




Gothic style, designed by him- 
self. Figs. 109, 110, 111, and 
112, represent some of the de- 
tails on a larger scale. 

Fig. 109 is an elevation of 
the bay window with a balco- 
ny over it, to the scale of one- 
qnarter of an inch to the foot. 
Fig. 110, the Tierge hoard of 
the small gable over this bal- 
cony. Fig. Ill, part of the 
verge board of the gable over 
the porch. Fig. 112 are chim- 
ney tops, snch as can be ob- 
tained of Garnldrk clay. 



Fiff. 109. 




Fi''. 110, 



PLATE LXYIIl. 



252 





AKCHITECTUKAL DRAWING. 



283 



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M 



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Cw 



ll 



m 03 






Fis. 11^ 




Fiff. 113. 



284 



AKCHITECTUEAL DRAWING. 




Plate LXIX. is the elevation and principal floor plan of a villa in the 
Italian style, as constructed after plans of Mr. Upjohn. "It is one of the 
most successful specimens of the Italian style in the 
United States." 

This villa is built of brick, jpainted externally 
of a light freestone color, and the window dressings, 
string courses, cornices, brackets, &c., are all free- 
stone. Figs. 113 and 114 are the front and side 
elevation of the balcony window in the front of the 
house, drawn to a scale of one-quarter of an inch 
to a foot. 

Figs. 115 and 116 are the first and second story 
plans, and Plate LXX., an elevation in perspec- 
tive of a tenant house, built by the 'New York 
Workingmen's Home Association, after designs by 
JohnW. Pitch, architect. 

The plans are admirably suited to the purpose 
designed, a separate house as it were for each 
tenant; sufiicient for the wants, and within the 
means of the laboring classes, especially adapted 
for the population of large cities. In its construc- 
tion it is almost entirely fire proof; the staircases 
are of iron, the hall floors are constructed with iron 
beams, brick arches, and blue flagging ; the divid- 
ing floors and walls are deafened ; every alternate 
wall is of brick ; every window has inside shutters, 
and every room is well ventilated, having air flues 
from each to the roof. 

In the lower story are large stores, with vaults in front. On the south 
is a spacious flagged court yard, 22 by 188 feet, wdiich is used by the in- 
mates for washing and drying their clothes — the families on each floor 
having the exclusive use of it for specified days of the week. The yard 
connects with the main hall, by cross halls, and is shut ofl" from the streets 
by high gates that are kept closed, except fuel is brought to the premises, 
when carts can be driven to each tenant's cellar, and at once deposit their 
loads. The cellar is divided into 04 compartments, that is, one for each 
tenement, with a lock and key for each. 

On the upper floor are two large adjoining rooms, measuring together 
53 by 50 feet, which can be tin-own into one, or disconnected at pleasure. 
They arc designed for lectures, concerts, or moral and educational uses for 




.-^j 



Fig. 114. 



PLATE LXIX. 



2S4 





PLATE LXX. 



2<44 







ARCITrrECTURAL DRAWING. 



285 



Fig. 115, 




286 



AECHITECTUKAL DEAWING. 



the inmates during tlie week, and for Sunday scliool and religious observ- 
ances on the Sabbath. 

The exterior is of brick with brown stone window-sills, and in its style 
Is an excellent example of the architectural effect that may be produced 
in our most common materials, and in an unpretending edifice, by break- 
ing up the monotony of fagade by even slight projections, by the clustered 
and circular heads of the windows, and by an appropriate and varied 
cornice. This style is becoming very popular and is particularly appli- 
cable to the construction of mills and workshops. 

Store and Warehouses. — Plate LXXl., is an elevation of a store front, 
and figs. 117 and 118 plans of first story and basement. 



mi n 



^^m^^^% 



^M 



B 



^Bnm^BBw 



These plans may be taken as a type of the general class of large whole- 
sale or retail stores covering but one lot. In this city there is usually 
beneath the sidewalks two stories, the basement and sub-cellar. These 
are generally let with the first story, and the upper stories together by 
themselves. The depth of the stores are mostly from 100 to 200 feet, on 
an average about 150 feet. The centre is lighted by a skylight in the roof, 
and by well-holes, B, beneath, in the several floors. In front of the en- 
trance is a platform. A, which is either an iron grating, or, when the base- 
ment extends through into the front vaults, covered with patent vault 
lights. To protect the vaults from moisture tlie walls are laid hollow, and 
the outside covered with asphalte. The lioistway to basement and sub- 
cellars, is by a trap in the gi-ating front of the window, usually a plat- 
form supported by chains at the four corners, and raised vertically, often 



PLATE LXXI 



2SC 













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__=. __^-„ „ .- 1 



ARCHITECTURAL DRAWING. 287 

by a car sliding on an incline, if tlierc are outisde stairs leading to the 
basement. In tlie rear an area of some ten to fifteen feet in width is dug 
out, and the two lower stories show full. All the rear windows are pro- 
tected bj iron shutters. 

The floor of the first stoiy is often laid with a rising grade, of about 
1 foot in 100 towards the rear, to prevent the appearance which a long 
level sometimes has of descending, and to afford more liglit in the rear to 
the basement. The offices are in the rear on this floor. Tlie safe is some- 
times built into the wall, or into a projection from it, or the safe is mova- 
ble ; or, what is rare at present, a book vault is made in the front vault. 
Tlie front windows and doors are mostly protected by revolving shutters 
rolling up like a curtain in the box lintels above. Separations are made 
between tenants occupying difierent floors by iron framed skylights over 
the w^ell-holes. 

C is the entry way to the second story, separated from the store by a 
glass partition protected by a wrought iron screen or guard. Above this 
entrance in the second floor, is the hoistway for goods, generally about five 
feet square. The second floor does not difl'er in -plsm from the first, and 
so with the stories above, except in some cases the well-holes are wider 
in the upper stories. The floors are all level. 

The water closets are mostly on the third floor, and in the front base- 
ment vault. The heating is either by stoves, hot air furnaces, or steam. 
The shelvings, counters and other furniture depend, of course, on the class 
and kind of business. 

JFront Elevation. — ^Various styles are adopted, but in one particular 
there is almost an uniformity ; that is, the whole front is suj)ported on posts 
of cast iron in the first story, with iron lintels and cornice ; the great ob- 
ject being to get as much light as possible in this story. These posts are 
sometimes square or rectangular in plan, with a small sunk ])^t\q\ on the 
face, and shield-like ornaments containing the number of the store, and 
capitals at the top ; sometimes a sort of Corinthian column, and some- 
times two posts, the inside one circular, and the outside square. As there 
is but little chance for ornament, the building seldom assumes any distinc- 
tive expression till it reaches the second story. The great ornament of 
the first story is the plate glass. Tlie elevation and plans represent the 
usual form of the wholesale stores with but'three openings in the first story 
— one window and two doors. In the retail stores occupying a full lot 
there are generally four openings, the door to the first floor, central between 
two windows, and the side door leading to the second story ; but where 
all the stories are occupied by the same trade, the side door is usually 



288 AKCHITECTUEAL DE AWING. 

omitted. The door of the retail store is generally recessed, with show 
windows at the sides to admit of the greater display of goods. The glass 
of the windows are sometimes of one plate, as large as 8 x 14 feet even, 
but more usually in four squares ; seldom more in number. 

Above the first story, the front begins to assume an architectural ex- 
pression, though seldom perhaps very significant of any intention or de- 
sign for a specific purpose inside. The example selected may be con- 
sidered a fair average of the class. It is to be remarked that where 
various businesses are to be carried on in the same building, and where 
large signs may be necessary to designate them, there will be but little 
room, as there will but little necessity, for much ornamental detail. 

Plate LXXII. is a chaste and beautifal fagade of two stores, erected 
on Broadway, from designs by J. B. Snook, architect. 

Plate LXXIII. is an elevation of a store front executed in cast iron by D. 
D. Badger & Co. of this city. The style is Yenetian, and when the front is 
more than fifty feet in width, the efi'ect is imposing. It is rather more ap- 
propriate for stores with ofiices above, or for stores designed for but one 
pur|)ose, as signs larger than could be placed in the panels would mar the 
efi'ect. Iron was first introduced for house fronts by Mr. Bogardus, and 
it has much to recommend it. Ornaments can be applied profusely, and 
at the same time cheaply, and in durability it exceeds our common free- 
stones. The chief objection at present lies in this, that few wish to go to 
the expense of new patterns : the result is that the forms become too ste- 
reotyped, especially objectionable when much ornament is used. The color 
which it should be painted has been a subject of much discussion ; the 
prevailing tint at present is a sort of cream color, with brown trimmings 
of the windows. 

School Houses. — ^Pl. LXXIY. contains a plan and elevation of a dis- 
trict school house, with seats for forty-eight scholars. There are two en- 
trances, one for each sex, with ample accommodations of entry or lobby 
room for the hanging up of hats, bonnets and cloaks. A side door leads 
from each entry into distinct yards, and an inside door opens into the 
school-room. The desk, T, of the teacher, is central between the doors, 
on a platform, P, raised some six or eight inches above the floor. In 
the rear of the teacher's desk is a closet or small room, for the use of 
the teacher. The seats are arranged two to each desk, with two allcj^s 
of eighteen inches, and a central one of two feet ; the passages around 
\\\Q room are three feet. The scale is ci<rht feet to the inch. The eleva- 
tion is in a vQiy phiin Eomanescpie style, to be constructed of brick with 
hollow walls. y 



?'J^TE,Ly.Xll 




ELEVATION 



■03FI.CO. OSBOnKES /"flOCfT.S.S 



IM.ATE LXXlll. 




-A// 



PLATE LXXIV. 



2S8 




AECHITECTUEAL DRAWING. 289 

On the Eequirements of a School- House. — Every scholar should have 
room enough to sit at ease, his seat should be of easy access, so that he 
may go to and fro, or be approached by the teacher without disturbing 
any one else. The seat and desk should be properly proportioned to each 
other and to the size of the scholar for whom it is intended. The seats as 
furnished by the different makers of school furniture, vary from nine to 
sixteen inches in height ; and the benches from seventeen to twenty-eight 
inches ; measuring on the side next the scholar. Tlie average width of the 
desk is about eighteen inches, and is formed with a slope of from one and 
a half to two and a half inches, with a small horizontal piece of from two 
to three inches at top. There is a shelf beneath for books, but it should 
not come within about three inches of the front. Tlie width of the seat 
varies from ten to fourteen inches, with a sloping back, like that of a chaii* ; 
it should, in fact, be a comfortable chair. It will be 
observed that, in the plate, two scholars occupy one I P I IQ I p fl 
bench ; fig. 119 represents another arrangement, in 
which each scholar has a distinct bench ; and, in h M p 
many respects it is preferable, but is not quite so — — 

economical in room. In pi'imary schools, desks are ] Ir-^ I Ir-j I V-^ [j 

not necessary; and in many of the intermediate I | | | | | U 

schools, the seat of one bench is formed against the i — i j — i i — i 
hack of the next bench ; but distinct seats are pre- LJ U |U 
ferable. Tlie teacher's seat is invariably on a raised Fig. 119. 

platform, and had better be against a dead wall than where there are win- 
dows. The best light is undoubtedly a skylight, but as this is seldom con- 
venient, the lights at the side should be high above the floor. Blackboards 
and maps should be placed along the walls. Care should be taken in the 
warming and ventilation ; the room should not be less than ten feet high ; 
the best method of heating is by furnaces in the cellar, warm air should 
be introduced in proportion to the number of scholars, and ventiducts 
should be formed to carry off the impure air. 

In cities the school-houses are made of a number of stories — the pri- 
maries being in the lower stories, and, in some cases, play rooms also, and 
the grammar schools occupying the whole floor above. In these cases the 
teachers are numerous, and separate rooms are prepared for the hearing 
of recitations. The style of interior finish should always be simple ; in 
the exterior various styles have been adopted. "We know of none more 
suitable than brick school-houses, with Romanesque or circular window- 
heads and coved cornices, as given in the plate, or similar to the Home, 
Plate LXX. 

19 



290 ARCHITECTUEAL DRAWING. 

Lecture Booms ^ Churches^ Theatres, Legislative Llalls. — To the proper 
constrnction of rooms or edifices adapted for these purposes some know- 
ledge of the general principles of aconstics, and their practical application, 
is necessary. In the case of lecture rooms and churches, the positions of 
the speaker and the audience are fixed ; in theatres, one portion of the 
enclosed space is devoted to numerous speakers, and the other to the 
audience ; in legislative halls, the speakers are scattered over the greater 
part of the space, and also form the audience. 

The transmission of sound is by vibrations, illustrated by the waves 
formed by a stone thrown into still water ; but direction may be given to 
sound, so that the transmission is not equally strong in every direction ; 
thus, Saunders found that a person reading at the centre of a circle of one 
hundred feet in diameter, in an open meadow, was heard most distinctly 

in front, not as well at the sides, but scarcely at 
^,- --., all behind. Fig. 120 shows the extreme dis- 

tance every way at which the voice could be 
distinctly heard : ninety-two feet in front, seventy- 
five feet on each side, and thirty-one feet in the 
rear. The waves of sound are subject to the 
same laws as those of light, the angles of reflec- 
tion are equal to those of incidence ; therefore, 
^ig- ^20. in every enclosed space, there are reflected sounds 

more or less distinct, according to the position of the hearer, and to the form 
and condition of the surfaces against which the waves of sound impinge. 
Thus, of all the sounds entering a parabolic sphere, the reflected sounds 
are collected at the focus. Solid bodies reflect sound, but draperies absoi-b 
it. As, in all rooms, the audience can never be concentrated at focal 
points, nor is it possible in any construction to make calculation for 
all positions, it is in general best to depend on nothing but the direct force 
of the voice, and not to construct larger than can be heard directly without 
aids from reflected sounds. 

There is great difference in the strength of voice of different speakers ; 
the limits as given in the figure are for ordinary reading in an open space. 
In enclosed spaces, owing to the reflected sounds or some other cause, there 
are certain pitches or keys peculiar to every room, and to speak with ease, 
the speaker must adapt his tone to those keys. The larger the room, the 
slower and more distinct should be the articulation. 

It has been observed, that the direction of the sound influences the ex- 
tent to wliich it may be lieard. Tlie direction of tlie currents of air through 
which the sound passes effects the transmission of the sound, and this may 




ARCHITECTUEAL DRAAVIXG. 291 

be made useful wlieii tlie rooms are heated by hot air, by introducing the 
air near the speaker, and placing the ventilators or educts at the outside 
of the rooms, and by placing their apertures rather nearer the bottom 
of the room than at the top. It would seem much better and easier 
to make a current of air a vehicle of sound rather than depend on re- 
flection. 

Tlie best form for a lecture room is the semicircle, or three-fifths of a 
circle, ^g. 120, the speaker in the one case at the centre, in the other, at 
the point A, on a platform raised some two or three steps above the floor, 
the audience being ranged in concentric seats, rising from the centre 
outwards. The room should be no higher than requisite for beauty 
or for ventilation. The ceiling should be slightly curved, not flat nor half 
globe. 

0)1 the space occiqnecl ly seats in general. — A convenient arm-chair oc- 
cupies about twenty inches square, the seat itself being about eighteen 
inches in depth, and the slope of the back two inches. Eighteen inches 
more affords ample space for passage in front of the sitter : this accommoda- 
tion would be ample. In the arrangement of seats at the Academy of Mu- 
sic the bottom turns up, and twenty-nine inches only is allowed for both seat 
and passage, and eighteen inches for the width of seat, which may be taken 
as the average allowance in width to each sitter in comfortable public 
rooms. In lecture rooms stalls are often used, the space there occupied by 
seat and passage being about two feet six inches. The alleys should 
be at the sides of the room, with two intermediate, dividing the seats 
into three equal benches, and not one in the centre, except in very 
large rooms, as the space thus left is the best for hearing and seeing the 
speaker. 

In the earlier churches, ceremonies and rites formed a very large part 
of tlie worship, the sight was ratlier appealed to than the hearing, and for 
this purpose, churches were constructed of immense size, and. with all the 
appliances of ornament and construction, witli pillars, vaults, groins and 
traceried windows. In the churches of this country, the great controlling 
principle in the construction of a church, is its adaptatton to the comfort- 
able hearing and seeing the preacher. In this view alone, the church is 
but a lecture room : but since even the character of the building may tend 
to devotional feelings in the audience, and since certain styles and forms 
of architectm-e have long been used for church edifices, and seem paj'ticu- 
larly adapted for this purpose, it has been the custom to follow these time- 
honored examples, adaj^ting them to the modern requirements of church 
worship. 



292 



AECHITECTUEAL DRAWING. 



Fig. 121, is a plan of an ancient basilican or Romanesque chnrcli ; 
fig. 122, a sectional elevation of the same. Fig. 123 is a plan of a GotMc 




church in which C is the chancel, usually at the eastern extremity, TT the 
transept, and ]^ the nave. In general elevation the Gothic and Roman- 
esque agree ; a high central nave and low side aisles. In the later Roman- 
esque the transept is also added. 

The basilicas aggregated within themselves all the offices of the Rom- 
ish church. The circular end or apex, and the raised platform, or dais in 
front of it, was appropriated entirely to the clergy ; beneath was the crypt 
or confessional where were placed the bodies of the saints and martyrs, 
and pulpits were placed in the nave, from which the services were said or 
sung by the inferior order of clergy. 

The plan, ^g. 123, is that of the original Latin cross, the eastern limb 
or chancel being the shortest, and the nave the longest. Sometimes the 
eastern limb was made equal to that of the transepts, sometimes even lon- 
ger, but never to exceed that of the nave. In the Greek cross all the 
limbs are equal. In most of the French Gothic churches the eastern end 
is made semicircular, often enclosed by three or more apsidal chapels, that 
is, semi-cylinders, surmounted by semi-domes. 

The Byzantine church consisted internally of a large square or rectan- 
gular chamber, surmounted in the centre by a dome, resting upon massive 
piers ; an apse was formed at the eastern end. Circular cliurches were 
built in the earlier ages for baptisteries, and for the tombs of samts and em- 
perors. 

Having thus briefly treated of the general form of ancient churches, 
we proceed now to the consideration how far they may be applied to 
the requirements of modern clmrch services. The prime necessities are 
those of tlie lecture room ; comfortable scats, convenient for hearing and 
seeing tlic preacher ; and proper provision for ventilation. In addition, an 
eligible position for the choir, a small withdrawing room for the clergy- 
man, and a room suitable for Sunday Schools and for parish meetings. 



AECHITECTURAL DRAWING. 203 

Seats are arranged by pews or stalls, the width of each pew being in gen- 
eral about two feet ten inches. Tlie length of pews is various, being gen- 
erally of two bizes, adapted to either small or large families, say from seven 
feet six inches, to eleven feet six, eighteen inches being allowed for each sit- 
ter. In arrangement it is always considered desirable that there should be a 
central aisle, and if but four rows of pews, two aisles against the wall ; if 
six rows, one row on each side will be wall pews. Few churches are now 
without an organ ; its dimensions should of course depend on the size of 
the church. In form it may be adapted somewhat to the place which may 
be appropriated to it. In general it is oblong in form, the longer side be- 
ing with the keys. The dimensions suited to a medium sized church are 
about nine feet by fifteen, and twelve feet in height. The withdrawing 
room for the clergyman may be but of very small dimensions, and should 
be accessible from without. The Sunday School, in general, requires in 
plan about half the area of the chm-ch. 

As city residences differ from those in the country, from the same ne- 
cessities do the city churches differ from the rural ones. A very common 
form of city church is, in plan, that of the Latin cross, with extremely 
short transepts and chancels ; sometimes the roof is supported by pillars, 
with imitated vaults in plaster, but often with a double pitch roof, and 
open timber finish in the inside. The organ loft is sometimes in one of 
the transepts, sometimes at the back of the congregation over the door of 
entrance. 

A sort of basilican church is also very common : rectangular in form 
with a small semicircular niche behind the preacher, and small withdraw- 
ing rooms or vestries at each side of it. The ceilings are finished after the 
Greek style, with sunk panels, sometimes coved, with pilasters but seldom 
pillars, except short ones, to support the galleries which are adopted in 
this style of buildings, but not so commonly in the Gothic. The rooms for 
Sunday Schools are almost invariably in the basement of the city churches. 

The basilican form is evidently the most economical in its occupation 
of land ; if the church be situated at the corner of two streets, it can cover 
the whole lot, one side, or a portion of one side being left blank of win- 
dows. If an elevation similar to fig. 122 be adopted, the light can be ta- 
ken in at clere-story windows. But this form is objectionable as requiring 
pillars in construction, which, unless made of iron, and of small size, very 
much interfere with sio^ht and hearino;. 

The position of our city churches is usually as we have said at the cor- 
ner of streets, but if they can be placed so far in the centre of a lot as to 
receive the light from the back areas, the position is preferable as removed 



294 A^CHITECTUEAL DRAWING. 

from the noise of passing veliicles. In that case the church proper is ap- 
proached by a long aisle, above which may be the room for the Sunday 
School. This room should be fitted with water-closets, in fact they would 
be often of great convenience connected with all churches. 

In elevation, city churches are Greek with porticoes in front, Eoman- 
esque and Gothic, occasionally Byzantine. The Greek have no tower but 
often a spire above the portico ; the Komanesque and Gothic generally one 
tower, over the central door of entrance, or at one corner ; sometimes two, 
one at each side of the principal door, almost invariably surmounted by 
spires, high and tapering, usually of wood, but in some instances of stone. 

Plate LXXY., is a design for a church in the English Decorated 
Gothic style. It will be observed in the design that there is a side en- 
trance with its appropriate gable ; in a similar way, small edifices may be 
attached to the main one, for necessary offices, parsonages, or Sunday 
schools, adding much to the picturesque effect, and particularly appropri- 
ate to country churches. 

Plate LXXyi. is the original design (by James Kenwick, architect) of 
the front elevation of the Eoman Catholic cathedral now being built in 
Fifth avenue. The style is the French Decorated Gothic, and the fagade 
is more extensive than that in process of construction. 

Plate LXXYII. is a design in the Eomanesque or Byzantine style, by 
Messrs. Eenwick & Sands, architects. It is now being built in Fourth 
avenue ; but the drawing is incomplete, being partially a working one, 
and a campanile forms a part of the design. 

It is the custom in Episcopal churches to place, if possible, the chancel 
at the eastern end, and often a large window at the extremity of the chan- 
cel. The light from this window should be very much subdued, as it comes 
full in the eyes of the congregation ; for the comfort of preacher and peo- 
ple a side or top-light in the chancel would be much better. A very beau- 
tiful effect is produced by skylight in the apse of the Eomanesque church, 
which, being high above the congregation, does not interfere with them, 
and affords the best light to lead the services. The light in churches 
should not be garish, but subdued and well diffused, which will be best 
effected by light from windows, placed as high up as possible. A single 
north window, in many small churches, would be sufficient for all purposes, 
would not injure the eyes of the congregation by cross lights, would add 
very much to the effect when the walls are painted in fresco or distemper, 
and, if suitable means are provided for summer ventilation, no other windows 
would be necessary. In some recent city churches the light in the daytime 
is taken entirely from skylights, and at night from gas lights placed in 



HATV. LXXY. 



294-. 




?LATE.LXX^■^ 



-^^^ 




n1 




AECniTECTUK^VL DRAWING. 295 

the roof of the church, and reflected below through the Scame apertures in 
the ceiling. 

Theatre, — ^The requirements of theatres and opera houses, differ essen- 
tially from those of lecture rooms and churches, in that the audience 
themselves form an important part of the exhibition. It is not only ne- 
cessary that the audience should have a good position for hearing and see- 
ing the performance upon the stage, but also to see cacli other. The most 
aj)proved form, now, for the body of the house, is a circular plan, the 
opening for the stage occupying from one-fourth to one-fifth of the circum- 
ference, the sides of the proscenium being short tangents. Tlie circular 
form is well adapted for both hearing and seeing, and also for lighting. 

In the general position of the stage, proscenium, orchestra, orchestra 
seats, parquette, and boxes, but one plan is followed. We proceed to give 
briefly the usual arrangements of seats, and some other requirements, and a 
small table of the proportions of different houses. The line of the front of 
the stage, at the foot lights, is generally slightly curved, with a sweep, say, 
equal to the depth of the stage, and the orchestra and parquette seats are 
arranged in circles concentric with it : of the space occupied by seats we have 
already spoken. The entrance to the parquette may be through the boxes, 
near the proscenium, and often centrally, but better at the sides, dividing the 
boxes into three equal benches ; the seats in the boxes are usually concentric 
with the walls, and more roomy than those of the parquette. The orches- 
tra seats are of a height to bring the shoulders of the sitter level with the 
floor of the stage, and the floor of the parquette rises to the outside, 1 in 
15 to 18. The floor of the first row of boxes is some 2 to 3 feet above the 
floor of the parquette at the front centre, and rises by steps at each row, 
some 4 inches ; in the next tier of boxes the steps are considerably more 
in height, and so on in the boxes above. In general, three rows of boxes 
are all that is necessary ; in front, above the second, the view of the stage 
is almost a bird's eye view. The floor of the stage descends to the foot- 
lights at the rate of about 1 in 50. In large theatres it is of the utmost 
importance that all the lobbies or entries should be spacious, and the 
means of exit numerous and ample. The staircases broad, in short flights 
and square landings, and not circular, as, in case of fright, the pressm-e of 
persons behind may precipitate those in front the whole length of the 
flight. Ladies' di-awing rooms should be placed convenient to the lobbies, 
of a size adapted to that of the theatre, arranged with water closets ; 
there should also be provided rooms for the reception of gentlemen's canes 
and umbrellas, with water closets attached. The box-office should be, of 
course, near the entrance, but so arranged as to interfere as little as possi- 



296 



ARCHITECTURAL DRAWING. 



ble with the approach to the doors of the house. At the entrance there 
should be a very spacious lobby, or hall, so that the audience may wait 
sheltered against the weather ; if possible, there should be a long portico 
over the sidewalk, to cover the approach to the carriages. But single en- 
trances are necessary to distinct parts of the house, but the greater the 
number of, and the more ample places for exit, at the conclusion of the 
piece, the better. 

COMPAEATIVE TABLE OF THE DIMENSI0I7S OE A FEW THEATEES. 



NAATK AND LOCATION. 




DISTANCE 


IN FEET 






HEIGHT, 


IN FEET. 


1. 

n 


_6>0.S 


It 


r 

C5 




Ii 
II 

« -J 
1-2 


Si 


1 


Alexandre, St, Petersburg, 


65 


11 


84 


58 


56 


75 


53 


58 


, Berlin, 


62 


16 


76 


51 


41 


92 


43 


47 


La Scala, Milan, 


77 


18 


78 


71 


49 


86 


60 


64 


San Carlo, Naples, .... 


77 


18 


74 


74 


52 


66 


81 


83 


Grand Theatre, Bordeaux, 


46 


10 


69 


47 


87 


80 


50 


57 


Salle Lepelletier, Paris, 


67 


9 


82 


66 


43 


78 


52 


66 


Co vent Garden, London, . 


66* 




55 


51 


32 


86 


54 




Drury Lane, " 


64* 




80 


56 


32 


48 


60 




Boston, Boston, 


53 


18 


68 




46 


87 


m 


58 


Academy of Music, New Tork, . 


74 


13 


71 


62 


48 


83 


74 




Burton's (New), " 


53 




40 


62 






52 


62 


Opera House, PhUadelpMa, 


61 


17 


72 


66 


48 


90 


64i 


74 


* These dlmensi 


ms include 


the distanc 


B between 


he foot-lig 


its and cur 


aiu. 







Although much has been written about the construction of legislative 
halls, in relation to acoustic principles, there yet seems to be great disa- 
greement in practical examples, and in the deductions of scientific men. 
The Chamber of French Deputies was constructed after a report of most 
celebrated architects, in a semicircular form, sm-mounted by a flat dome, 
but as the member invariably addresses the house from the tribune, at the 
centre, in its requirements it is but a lecture room. Mr. Mills, Architect, 
of Philadelpliia, recommends for legislative or forensic debate, a room cir- 
cular in its plan, with a very slightly concave ceiling. Dr. Eeid, on the 
contrary, in reference to the Houses of Parliament, gave preference to the 
square form, with a low, arched ceiling. The Hall of Eepresentatives nearly 
completed at Washington, is 139 feet long, by 93 feet wide, and about 36 
feet high, with a s])acious retiring gallery on three sides, and a reporter's 



29T 



PLATE LXXYIII. 




AKCHITECTURAL DRAWING. 297 

gallery behind tlie Speaker's cliair. The members' desks are arranged in a 
semicircular form. The ceiling is flat, with deep sunk panels, openings for 
ventilation, and glazed apertures for the admission of light. The ventila- 
tion is intended, in a measure, to assist the phonetic capacity of the Hall, 
the air being forced in at the ceiling and drawn out at the bottom. 

In reviewing the general principles of acoustics, it will be found that 
those rooms are the best for hearing in which the sound arrives directly to 
the ear, without reflection ; that the sides of the room should not be re- 
flectors, not sounding boards, and that surfaces absorbing sound are less 
injurious than those that reflect. Slight projections, such as ornaments of 
the cornices and shallow pilasters, tend to destroy sound, but deep alcoves 
and recessed rooms produce echoes. Let the ceiling be as low as possible, 
and slightly arched or domed; all large external openings should be 
closed ; as M. Meynedier expresses it, in his description of an opera house, 
" Let the hall devour the sound ; as it is born there, let it die there." 

Crystal Palaces. — PL LXXYIII. represents a perspective view of the 
interior of the Xew York Crystal Palace, figs. 125, 126 a half plan of the 
ground floor and of the galleries. This building originated from that of 
London, of which some of the details of consti'uction have been already 
given. These buildings are composed wholly of iron, p;lass and wood, but 
no large pieces of either material are used ; in this consists their great pecu- 
liarity. Stiffness and tenacity of material are applied rather than mass, to 
counteract incidental strains ; and, on this account, they are not as suitable 
as walls of brick and stone for permanent structm-es, nor are they as cheap ; 
and in this respect, an improvement has been made in the French exhi- 
bition building ; but for a structure easily moved and put together, as it was 
intended, and for green or hot-houses, it seems especially adapted ; and as 
a practical example of the application of iron, and an economical applica- 
tion, it has been of great importance. 

Before concluding the article on architectural drawing it may be ap- 
propriate to speak briefly of materials as applied in tlie exteriors of edifi- 
ces. Sufficient has already been said of their strength, we now refer 
merely to their fitness to architectural ornament. 

Brick in cities is by far the most common of all materials, nor do we 
know of any more suited to workshops and factories, for appropriateness, 
economy, and durability (when hard burned), nor do we know of any style 
of architecture more fitted to the material than the Pomanesque, as in 
Plate LXX. Stone in the rough or rubble walls, laid in cement or mortar, 
are often used for these structures, but in that case the lintels should be 
square, and it possible of a different shade of stone. 



298 



AECHITECTUKAL DEAWING. 




AKCniTECTTRAL DRAWING. 299 

For city residences, and stores, the exteriors are composed of all sorts 
of building materials, with the exception of wood, from its insecm-ity in 
case of fire ; brick, with marble, freestone, iron or terra cotta lintels and 
sills for openings, red brick and straw-colored bricks, brick on rusticated 
basements, and sometimes brick in alternate stripes with marbles ; free- 
stone in a great variety of shades, mostly of a reddish brown, often fawn 
and drab ; marbles white and veined ; native and foreign granite ; and iron, 
the use of which in fronts is the invention of our age, and is destined to 
modify our style of architectm-e. 

All materials are suited for country residences except iron; stone 
houses may be kept in their native color, but brick or wood should be 
painted. TTe extract from Downing the following on the color of country 
houses. ""\re think all buildings in the countiy should be of those soft 
quiet shades called neutral tints, such as fawn, drab, gray, brown, etc., and 
that all positive colors, such as white, yellow, red, blue and black should 
always be avoided ; neutral tints harmonizing best with nature and, posi- 
tive colors most discordant. 

In the second place, we would adapt the shade of color as far as possible, 
to the expression, style or character of the house itself. A large mansion 
may receive a somewhat sober, dignified hue ; a house of moderate size, a 
lighter and more pleasant tone ; small cottages should always have a cheer- 
ful, lively tint, not much removed from white. Country houses thickly 
surrounded by trees, should always be of a lighter shade than those stand- 
ing exposed. In proportion as a house is exposed to view, let its hue be 
darker; and where it is much concealed by foliage, a very light shade of 
color is to be preferred. 

" A species of monotony is produced by using the same neutral tint 
for every part of the exterior of a country house. A certain sprightliness 
is bestowed on a building in neutral tint by painting the bolder projecting 
features of a different shade. The simplest practical rule that we can sug- 
gest for effecting this in the most satisfactory manner, is the following: if 
the tint selected for the body of the house be a light one, let the facings 
of the windows, cornices, etc., be painted several shades darker of the 
same color. The blinds may either be a still darker shade than the fa- 
cings, or else the darkest green. If on the other hand, the tint chosen is 
a dark one, then let the window dressings, etc., be painted of a much 
lighter shade of the same color.'' 

Thus far Mr. Downing. Most persons must be struck with the justness 
of his remarks in general, but all are not prepared entirely to ignore white 
as a color for country houses. "We have always fancied in contemplating 



300 AKCHITECTUKAL DRAWING. 

an extensive landscape tliat jottings of white enlivened the scene, and 
prefer a whitewashed cottage, carrying an air of cleanliness, to the least 
admixture of neutral tint : neither seems it high art to harmonize always 
with nature, it often makes a very flat picture. 

However we build, or whatever built of, let the building express the 
purpose, and let the material be suited to it. Let those which are intend- 
ed for time be of lasting materials, but those that are temporary, be of 
that most convenient ; let not one imitate the other. 

Yentilation and Warming. — To the proper construction of all edifices 
some knowledge of the principles of ventilation and warming are neces- 
sary, as the arrangements for this purpose are to be made in planning the 
building. Air is deteriorated in apartments by the respiration and perspi- 
ration of people, and by combustion in heating and lighting. At least 3 
cubic feet per minute of fresh air should be supplied for each person occu- 
pying the room, this quantity being deteriorated by respiration and per- 
spiration. As to combustion, 1 pound of carbon or charcoal, in burning, 
consumes 2.6 pounds of oxygen, which is that contained in between 13 
and 14 pounds of atmospheric air ; and 1 pound of hydrogen, consumes 8 
pounds of oxygen, which is that contained in about 40 pounds of atmos- 
pheric air. l^ow tallow, wax and oil contain upon the average from 77 to 
80 per cent of carbon, and from 11 to 14 per cent, of hydrogen : the per 
centage of carbon in anthracite and bituminous coal is more various, but the 
same calculations may be used. 100 cubic feet of air weighs about 7 
pounds, so from the above data the approximate consumption of oxygen 
by any given quantity of the above combustibles, is easily calculated. 
The combustion of coal gas generally spoils thrice its bulk of oxygen, or, 
fifteen times that of air. 

The methods of warming most generally practised in this country 
are by hot air furnaces ; in which coal is consumed, in furnaces inclosed 
within a brick chamber into which the fresh air is introduced from out of 
doors, heated, and conveyed usually by tin conductors to the various 
rooms of the building. In this case it will not of course be necessary to 
find how much air is required for the combustion of the fuel, as the air 
for this purpose is introduced from the cellar : it is only requisite to deter- 
mine how much may be deteriorated by persons occupying the rooms, and 
how much by the lighting. Having determined this quantity, we make 
provision for introducing the amount through a cold air box to the hot air 
chamber of the furnace. The velocity of the current in this box, to deter- 
mine the size of the box, wo call 4 feet per second. This box is provid- 
ed with a slide valve, to regulate the amount of air furnished ; in extremely 



AKCIIITECTURAL DKAWIXG. 301 

cold weather tlie air may pass too rapidly through the chamber without 
becoming siilficiently heated. The size of the conductors is less in propor- 
tionate area than that of the air box, usually not more than half, depend- 
ing somewhat on the vertical length of the conductor. The higher the 
conductor the stronger the current, and the less the necessary area. All 
hot air flues should be removed at least two inches from wood work. Hav- 
ing provided means for the introduction of air, it is necessary also to pro- 
vide means of egress. In general the cracks of doors and windows provide 
some little outlet, but hardly adequate to the requirements of j^ublic 
rooms. There should be ventilating flues, somewhat larger than the hot 
air flues. It is the general practice to introduce the hot air into the room 
at or near the bottom, and there is considerable disagreement where the 
openings of the ventilators should be, whether at top and bottom, or top 
only or bottom only. In the English Houses ot Parliament, the hot air is 
introduced at the floor, and the ventilating flues are in the ceiling ; in our 
House of Eepresentatives exactly the opposite course is to be pursued, the 
hot air is forced in by a fan into the top of the room, and taken out at 
the bottom ; probably as long as sufficient air can be got into and foul air 
out of the room, it matters little whether it is introduced at top or bot- 
tom. It is evident that in common rooms, when the current is not influ- 
enced by the ventilating arrangement, the nitrogen of the vitiated air 
rises to the ceiling, whilst the carbonic acid falls to the bottom, and as the 
former is more in quantity, the uppermost stratum of air is the foulest; and 
in our view, if the foul air be drawn ofl* at a height but little above the 
height of the persons in the room, it would seem sufficient for ventila- 
tion, and if the fresh air be introduced into one side of the room, and the 
foul air taken out at this height on the other, there would be a warm cur- 
rent of air circulating at the height most efi*ectual for warming the occu- 
pants of the room. 

All ventilating flues should be provided with valves or regulators ; thus 
when it is not necessary to change the air, by retaining it the heat is retained, 
and if the cold air from the room can be supplied to the air chamber of 
the furnace so as to establish a current, the heat will be much economized. 
The previous remarks on ventilation belong more appropriately to the 
heating and ventilating of public edifices, and rooms occupied by numbers 
of people ; in private houses, in general, there are but few, and the amount 
of air deteriorated but small, and if heated with open fire-places and 
grates, the cracks of the doors and windows supply plenty of air for venti- 
lation ; but the objection to this are the draughts of cold air. Tlie most 
perfect way for heating private houses is by a small furnace or hall stove. 



302 AECHITECTUEAL DRAWING. 

taking a supjDly of fresh air from outside and warming the halls, and open 
fires in the occupied rooms. In a sanitary point of view no arrangement 
of ventilation and warming can supply the place of a radiant fire ; but in 
view of the inconvenience of numerous fires, the next best thing is heat- 
ing by low steam, or by hot water pipes, a coil being placed in air cham- 
bers in the cellar, with a fresh air supply, and coils in all the necessary 
rooms. ISTo room in a house should be without a flue, or movable fan- 
lights over the door for ventilation, and if an air tube can be carried into 
a flue which is always heated, a current is secured. Steam is now used 
extensively to heat factories and workshops, and is decidedly the most 
cleanly. The general arrangement is in rows of | or 1 inch pipe against 
the walls of the room ; one foot in length of | inch pipe being the length 
considered adequate to heat fifty cubic feet of sj)ace. If there are many 
windows in the room, as they are very cooling surfaces, and also very sel- 
dom tight, more length of pipe should be allowed. Steam is used at va- 
rious pressures, but low or exhaust steam is preferable, as the pleasantest 
to the occupant, being in this respect like hot water. The desideratum 
in all warming apparatus is to heat with a surface not exceeding the heat 
of boiling water. The objection to water lies in its expense, and the dan- 
ger if not kept constantly at work in the winter of freezing. It is not 
therefore adapted to places which are to be heated intermitingly. 

Yentilators. — Although flues may be made for ventilation, still it is 
not always certain that there will be an ascending current ; often chimneys 
draw but poorly. In public buildings an artificial draught is created by a 
fire or by a fan, as has been done in the English House of Parliament. 
The usual expedient in this country, is by some of the common cowls for 
smoky chimneys, but the best appears to be Emerson's Yentilator. 

Much has been written on the subject of ventilation and warming, and 
many expedients, undoubtedly adequate in themselves, have failed, from 
the carelessness of servants and from want of attention. The grand requi- 
site seems to be, something that will be sure, and will not get out of order. 
It is now but a half a century since gas was introduced for lighting ; it has 
now been applied for cooking and warming, but not to a large extent eco- 
nomically ; whether it may be brought into general use for this purpose 
is a problem yet to be solved ; but steam, as now applied in most ISTew 
England factories, from a central set of boilers, could easily be applied to 
the warming and ventilation of many houses, and for many culinary pur- 
poses, and with gas would supply all requirements. 



AKCHITECTTJKAL DRAWING. 303 



SPECIFICATIONS. 

The following blank Specifications for Mason's and Cai-penter's work 
are intended as illustrations of the usual forms of such papers. 



Specification for Mason work. To be performed in building, erecting 
and finishing a Dwelling House ; to be situated on , between , 

for . Agreeably to drawings made and prepared by , Archi- 

tects. 

Date. 

Size of Building. — 

Excavation. — Dig out and cart away all the ground (that may be necessary), for Basement, 
Cellar, Areas, Drains, Cesspools, and footings for foundations, and cart away all 
the superfluous rubbish that may be made during the progress and at the comple- 
tion of the building. 

Materials and Worl:manship. — AU the several materials used in or about this Building, 
are to be of the very best quality, and all the work to be done in the best and most 
workmanlike and substantial manner, under the direction and to the satisfaction of 
the architects. 

ft. in. 

Height of Stories. — Under Cellar, 

Basement, , 

First Story, 



AU to be in the clear between floors and 
ceihnffs when finished. 



The roof to pitch 



Base Courses. — All the base courses to be of the best stones, and to be .... wide lo 

aU the walls and piers throughout, and all to be .... thick, laid close and solid, 
and of parallel thickness. 

Blue Stone Walls. — Build up all the stone walls agreeably to the di'awings and figures 

thereon, with the best quality of stones (in cement), properly hammered square 

and straight, laid close and level, well bonded and flushed in. 

The face of all the stone walls that are in sight, are to be neatly pointed. 



304: AKCHITECTURAL DRAWING. 

All the stone walls that come in contact with the ground, are to be properly- 
coated on the outside with Hydraulic Cement. 

Concrete. — The whole of the cellar is to be concreted with broken stones and Hydraulic 
lime laid .... thick in the very best manner. 

Coal Slide. — ^Build a coal slide, as per plan, and cover the same, both top and bottom with 
flag stones, well laid and bedded in cement, and to a proper grade. 

Brick Wbrh — ^Build up all the brick walls agreeably to the drawings and figures thereon. 

The chimney of brests fire places in Kitchen and Laundry, and chimney stacks 

above roof, and all the rear front, are to be faced with the best quality of 

pressed bricks of uniform deep red color. All laid in running bond and white putty 
mortar, and tuck-jointed. 

All the residue of the brick work throughout the premises to be done with the 
best, well burnt, hard, .... bricks, all the brick walls in Basement and Areas to be 
built in cement. 

Build Piers and Arches in cellar, and Piers and Arches under hearths, and build 
proper Ash-pits as shown on plan. Build all necessary flues, turn trimmer-arches 
to aU the fireplaces, arches to all the openings of the width of the openings. Parge 
the flues and scrape the same at the completion. 

Build up the hot air pipes as shall be directed. Leave indents in walls for hot 
air tubes, &c. The side walls are to be carried up two courses of brick above the 
roof planking to receive the tin, and then two courses on the tin, laid in cement, 
to receive the coping. 

All the fireplaces are to have brick inner hearths. All brick walls that come 
in contact with the ground, are to be well cemented on the outside. 

Cut away for, and make good after Plumbers, Purnacemen, Gasmen, &c., and 
finish all the brick work complete in the best manner. 
All the front plain Ashlar is to be backed in with wall built in ... . 

Drain. — ^Build a drain, to sewer on .... street, in the best manner, and all to be properly 
cemented ; also make all necessary blind drains, and put in proper traps, &c., com- 
plete, and cover the drain with flagging, and all to be done in the best manner and 
as shall be directed. 

Build cesspools, .... diameter and .... deep, in 4 inch brick work as shall 
be directed. 

Flagging. — All the flagging to be done with the very best quality of flagging, and 

not to be less than thick and .... super., on the face, and to be well smoothed 

ofi". 

Flag all the lay and bed in the concrete, and join neatly and perfectly tight. 

ring the sidewalk on .... with large sized strong flagging, .... thick, squared 
and fine axed, out of wind, and the joints laid solid in mortar, and the walk to be 
.... stones in width — have blue cut stone curbs, .... by ...., and cut gutters 
.... by .... agreeable to city ordinances. 



ARCHITECTURAL DRAWING. 305 

Lay the hearths in Kitchen and Laundry with blue stone flagging, as shown on 
plans, to be .... wide and .... thick, and polished. 

CUT-STONE WORK. 

All the Cut-Stone work is to be done with the very best quality of Stone, of 
close, fine grain and of uniform color. Cut and set in the very best manner, with 
oil putty, and agreeably to working di-awings, pointed and cleaned off, and all to 
be polished except what is mentioned to be tooled. 

The Chimney Stacks and Flues are to have caps as per elevation, cramped and 
leaded. 

Cope the wall to both sides of main buildmg with tooled coping, 3 inches by 14 
inches, cramped and leaded, and bedded in cement. 

All the windows to rear of building to have sills by .... washed and 

throated and moulded lintels by to doors and windows. Sills to door- 
ways by Water table to be by Copings to gratings by 

Have jambs and mantels to Kitchen and Laundry fireplaces : jambs by 

. . . . , cut in the usual way. 

Front Worlc. — ^All the windows to front on to have architraves, pilasters, and 

trusses and cornice, as shown on elevation — moulded sill .... by ... .and blocks 
under. 

Provide and put up the rusticated ashlar to first story .... thick, and the cor- 
nices, panels and balusters over a moulded water table to first story, and balus- 
ters and moulded sill course supported by heavy consols. 

The rusticated ashlar to be in courses, as shown on drawings, not less than 

bed, and .... reveals to windows, and cramped, resting upon moulded sill course. 
The plain and rusticated ashlar to return .... in the doorways, and all the front 

area walls and under stoop throughout to be faced with ashlar, and to average 

bed. Coping to area walls by Coping to receive railing .... by .... 

and moulded. Moulded steps by .... Sills to basement doorways to be 

by.... 

All the front on , above first story, is to have plain ashlar, and not to be 

less than .... inch bed, properly cramped with two cramps to each stone, and all 
the architraves to all the windows are to have .... inch reveals. 

Front Steps and Door Piece. — Put up the steps and door piece as per drawings and working 
drawings, moulded steps, and moulded solid strings, and panelled ashlar both sides, 
and pedestals, rails and balusters complete. 

Provide and put up the cornice properly cramped and anchored, and cut grooves 
in top for copper. 

Iron Worlc. — ^Provide a sufficiency of anchors, ties, and cramps ; and anchors for cut-stone 

work, and iron rims and covers to flues for stove pipes, and registers in kitchen, 

laundry and chimney brest; have gratings to areas as per plans, one grating to 

be made to fold open with chain, fastenings, and gratings to small cesspools. Pro- 

20 



306 ARCHITECTURAL DRAWING. 

vide proper plate, chain, and fastenings to Coal Slide. Provide and set wrought- 
iron guard bars to all the basement windows in rear. 

All the residue of the windows in basement to front on are to have or- 
namental wrought-iron panels to fill up the whole size of the windows. 

Provide and put up, strong iron lintels to all openings in walls shown on the 
plans. Provide and fix iron doors and frames to ash-pits and flues in cellar. 

Finish and paint all the iron work twice over ; brown stone color and sanded. 

Plastering. — All the lathing to be done with narrow laths, free from sap, reversing the 
heading joints every 18 inches. 

The mortar for the two first coats to be made up with the very best quality of 
.... lime and sharp grit sand, and the best long fresh cattle hair. 

All the hard finishing work to be done with the best lump .... lime and pure 
white marble dust, all the works to be properly gauged with plaster of Paris. 

Lath, scratch-coat, brown and hard finish, all the walls, ceilings, partitions, studs, 
and furring of every description, throughout the building, except walls of under 
cellar, which will be whitewashed twice over. 

The browning to be made plumb and straight, and to be properly hand floated, 
the hard finishing to be trowelled hard and smooth in the best manner. The ceil- 
ings of cellar to have one thick coat, and hard finished. 

Cornices. — The bed, bath and dressing rooms, to have a moulded cornice to girt .... 
inches. 

The drawing room, dining room, library, hall, and vestibule on first story, to 
have a large moulded and cove cornice with one row of large enrichments. 

The reception room ceiling to be groined, and put up large moulded cornice, fin- 
ished plain for painting, the niches to be finished with a rule joint. 

Form a dome over centre of staircase, in panels, with enriched mouldings. All 
the ceilings of principal story, are to be formed out in various panels, and large 
raised enriched mouldings. 

Put up a centre piece in hall, diameter, and one . . . c, and 

Working drawings at large will be given for all cornices, &c. Make good all 
damages during the progress and at the completion of the work. And should 
any blisters, cracks or defects appear during the progress or within six months after 
the completion of the building, the same is to be rectified without any extra chai-ge. 

story floors, are to be deafened with the best quality of deaf- 
ening. 

Provide materials for setting all the mantels, and all the grates. (The grates and 
mantels will be furnished by the owner.) 

Make good after plumbers, bellhangers, gasmen, and other works necessary to 
be done in the mason's department. And finish all the several works complete to 
the true meaning and intention of the drawings and this specification, leaving all 
the works sound, clean and perfect, at the completion of the building ; and the 
building is to be properly cleaned out at any and all times when directed by the 
architects, and if neglected by the contractor, the architect shall have the power to 
have it cleaned out, and the expense charged to the contractor. 



AKCHITECTURAL DRAWING. 307 



Specification for Carpenters' Work. To be pei*formed in building, 

erecting, and finishing a dwelling house, to be situated on , for 

Agreeably to the drawings made by Architects. 

Date. 
Size of Building. — 

Materials and Workmanship. — All the several materials used in the erection and finishing 
of this building are to be of the very best quality. Timber to be -well seasoned 
white pine free from all defects. All the lumber for joiners' work, to be clear white 
pine, and well seasoned, and free from all defects, and all the works throughout to 
be done in the best, most workmanlike, and substantial manner, under the direction 
and to the satisfaction of the architects. 



Height of Stories. 



Floor and Eoof Beams. — ^Basement beams .... by . . . ., and from centres. First, 

...... story beams, .... by , and .... from centres. Roof beams, .... 

by .... from centres. 

All the trimmers and headers to be thick, by the depth of the beams in 

each tier. 

All the beams are to be properly framed and wedged, for fireplaces, staircases, 
skylights, cfcc, and to be well laid solid on the walls. Xo blocking will be 
allowed. 

Have .... rows of double cross bridging to each tier of beams, and springing 
pieces for arches, and strong anchor strips. 

The ceilings of basement, and stories are to be cross furred with inch fur- 
ring 2 inches wide, and to be 12 inches from centres. The basement, story 

floors to be prepared for deafening. 

Partitions. — Put up all the partitions throughout, agreeably to the drawings, all to be 
properly framed and bridged in the strongest manner, as shall be directed. AH the 

partitions are to have door studs and heads by , sills and plates by 

, filling in studs by and from centres ; all to have upright block- 
ings and templates. 

Furrings, Wall Strips, &c. — Stud out for boxing and sliding shutters, niches and circular 

corners, with studs by , and from centres ; fur out all the walls with 

inch furring 2 inches wide and 12 inches from centres. 

Put up lintels to all the openings thick by their respective widths, pro- 
vide a suflBciency of yellow pine planks, wall strips, plates and wood blocks, and such 
other timber that may be required to complete aU the furrings as shall be directed. 



308 ARCHITECTTTEAL DEAWma. 

Dome and Ceiling Joist, — Form the dome over principal staircase witli and framing 

as shall be directed, bracketed out for panels, cornices, &c. 

The reception room ceiling is to be groined, and form a square dome and soflQts. 

Provide all necessary brackets and furrings for circular soffits, and cradlings for 
plasterers, centres for arches, and turning pieces for apertures ; all the centres to 
be made in a strong and substantial manner. 

Boofs and Slylights. — All the roof is to be covered with merchantable planed, grooved, 
and tongued planks, free from large or loose knots and other defects, laid in courses, 
and form proper gutters as shall be directed. Provide and put up an iron scuttle to 

roofs by , hung with strong, wrought straps, hinges, and iron curbs; put 

on chain, hasp, staple and padlock. 

provide and fix an iron hipped skylight over by . . . . , with proper cappmg 

and ventilation, worked with pulleys, weights, and cords. Provide and fix a hori- 
zontal skylight in dome as shall be directed. 

Floorings.— L&j the basement floor with the best quality 1^ inch Yellow Georgia pine, nar- 
row planks, and the , story floors, with the best quality of narrow, clean, 

1\ inch white pine planks, not to exceed 3^ inches wide, all to be planed, grooved, 
and tongued, laid in courses, and blind nailed. All the floors are to be neatly 
cleaned off at the completion, put borders to hearths, and saddles to doorways. 

Sashes and Frames.— A:^ the sashes and frames are to be made agreeably to drawmgs, all 

the windows to the are to have French casements, hung with butts, and 

sash fastenings. All the residue of the windows are to have ovolo sashes, 

double hung with cords, axle pulleys, weights, and box frames, sunk sills, 

sash fastenings. All the sashes in partitions are to be made, hung and finished, 
as shall be directed. All the rear sashes are to be made imitation French. 

Put up rough grounds for all bases, doorways, and windows, throughout, for ma- 
son to finish plastering to. 



ATTIC. 

Bases.— All the rooms, entries and landings, are to have 1^ mch moulded bases, . . . inches 
high. Closets to have inch headed bases, inches high. 

Windows.— AR the windows are to have li inch framed and moulded backs, plain linings, 
and moulded architraves. 

Doors.— AW the doors are to be inches thick, four panel, moulded both sides, hung 

with inch butts, and city-made mortise locks, single rebated jambs, and ar- 
chitraves as described to windows. And put up in all the closets iron hooks, and 
shelves in each, and beaded rails. All the bedroom doors are to have fan lights and 
to swing on a pivot, and put up the large closets and trimkroom as shall be direct- 
ed by the owner, with wide shelves and iron pins. 



ARCHITECTURAL DRAWING. 309 



THIRD STORY. 



Bases. — . . 
Windows.- 
Doors. — . . 
Closets. — , 



SECOND STORY. 



Bases. — All the rooms, entry, and landing, througliout, are to have . . incli sunk plinth, . . 

inches high, and moulded base to girt , and the plinths to be grooved and 

tongued into the floors. 

Windows. — All the windows on this story are to have boxing and sliding blinds, and the 
architraves .... 

Doors. — All the doors on this story are to be thick, panelled and raised mouldings 

both sides, hung with .... wide butts and mortise locks — 1^ inch rebatted 

jambs and architraves, as described to windows. 

The sliding doors are to have brassways, shieves, stops, and mortice locks, 

complete, and double set of farniture. 

Closets. — The closet doors are to be All to be fitted with jambs and narrow 

moulded architraves inside. 

Fit up all the closets with drawers, shelves, locks, knobs, and folding doorways, 
large brass hanging pins and porcelain heads, complete, to suit the owner. 

All the wide jambs and soffits to have moulded panels to match the doors. 

All the closets in dressing rooms, private room, to be fitted up with 

Water &tosets and Baths. — ^Fit up the water closet with red cedar seats, risers, and clamped 
flaps, hung with 3 inch brass butts, and 1]- inch dovetailed cistern; and put up 
moulded panel framing to baths, and put up pilasters, caps, cornice, and arches 
complete. 

Fit up all the wash stands with moulded and bead and butt dwarf panel doors and 
hinges, knob and brass spring catches, and shelves complete, as shall be directed. 

PRINCIPAL STORY. 



Windows. — 

All the window shutters to rear to be lined with iron in the neatest manner. 
The sliding shutters to rear windows in library to be .... thick, in two 



310 ARCHITECTURAL DRAWING. 

thicknesses, panelled and moulded both sides, with iron in between, and have proper 
brass ways, shieves, astragals, stops, bolts, and mortice latches with flush handles, 
complete. The bay window to have inside folding blinds, hung with strong hinges, 
bolts and fastenings, and to fold back into a box. 

Doors. — All the folding doors to be .... thick, and raised mouldings both sides, hung 
with 6|- inch wide plated butts, 6 inch city made mortice locks and large furniture, 
and plated flush bolts to the vestibule doors. 

The sliding doors to be .... thick, panelled, and mouldings both sides, and to 
have brass ways, . . inch shieves, double astragals and stops ; mortice locks, flush 
crank handles, and double set of furniture, and all plated. 

All the doors to have . . inch double rebated, moulded and panelled jambs ; 
moulded .... architraves wide, and plinth blocks. 



Closets, — . 



Wainscotting. — The dining room and library are to be wainscotted all around with panelled 
framing, and moulded capping, 3 feet 6 inches high. 

Dumb Waiter. — ^Put up a dumb waiter from basement to first story, with all its apparatus 
complete, of the latest and most approved construction ; with dwarf panel moulded 
doors, locks and knobs complete. And the butler's pantry is to be fitted up with 
pilasters, caps, cornice, and arches complete in the neatest manner. 



BASEMENT. 

Bases. — 

Windows. — 

Doors. — 

The doors leading from the laundry to the yard, and also from laundry to kit- 
chen, and from kitchen to hall and pantry, and the front entrance doors in basement, 
to be made sash doors, and to have proper bolts and mortice locks ; and sliding doors 
to rear to be lined with iron, with shieves, way stops, bolts, mortice locks and rings 
complete. 

The sashes in the front entrance door in basement are to be hung, and put in 
iron lattice work the size of the panel. 

Closets and Pantries. — 

The kitchen and laundry walls and partitions all around, will be lined with nar- 
row clear boards, planed and grooved tongued and beaded, 2 feet 10 inches high 
and properly capped. 

Fit up the laundry with wash tubs made perfectly water tight, with lids com- 
plete. 

Fit up the water closets and wash stands. 



AECniTECTURAL DKAWIXG. oil 

Under Cellar. — All the openings in cellar are to have 1^ inch bead and butt folding sash 
doors, proper frames hung with 4 inch butts, and thumb latches, and proper bolts. 
Cut awaj, and make good after plumbers, and case all the pipes. 
Fit up the cellar with cool bins, &c., as shall be directed by the owner. 

STAIRCASES. 

Put up the principal staircase agreeably to drawings, 1^ inch .... treads, 
moulded and returned noseings, 1| inch rises; moulded, carved and panelled front 
strings, .... sunk wall strings, all to be housed and wedged, put upon strong car- 
riages, and rough bracketed. Have .... moulded and double beaded hand rails, 

and newels, and balusters, all to be of the best ; the balusters and strings to 

be varnished .... times over, and the rails and newels to be polished. 

Put up the other staircases of white pine, with proper treads, risers, rails and 
balusters complete in the usual way ; as shall be directed. 



Bay Window. — 



Entrance Doorway. — Put up the principal entrance doorway as per elevation, of 

the door to be made folding, and of two thicknesses ; moulded both sides, hung 

and lock, and two mortice bolts. 

Cornice. — Put up a handsome block cornice to rear of building, as per working drawings, 
put up in the best manner. 

Working drawings at large will be given for all moulded works. 
Furniture. — All the knobs of every description throughout the . . . . , to be .... 

All the residue of the furniture and knobs throughout the building, to be of .... 

All the locks and fastenings throughout the building to be the best manufacture. 

SpeaTcing Tubes. — Provide and put up .... speaking tubes, from .... to . . . ., with .... 
mouth-pieces and covers. 

Tin WorTc. — Tin the roof with the best quality of single cross tin, properly soldered. Put 
up leaders . . inches diameter, made of strong double cross tin, and have 6 inch 
strong copper socket pipe from gutters, and globe basket cover. The lower length 
of the leader to be of cast iron, with a shoe. 

Scrape off all the rosin, and paint all the tin work three times over with the best 
white lead and oil. 

The top of cornices and gutters are to be lined with . . ounce copper. 

Bells. — Provide and hang . . pulls of bells, with strong copper wire and tin tubes, all to 
have lever chain pulls and proper cranks ; . . pulls to be plated, and all the resi- 
due to be of porcelain. 

Blinds. — The rear windows are to have outside Venetian blinds, moulded both sides, and 
painted French green, hung with strong hinges and patent fastenings and catches. 



312 AECHITECTUEAL DEAWING. 

Painting. — Paint all tlie woodwork throughout, "both exterior and interior, three times 
over in the hest manner, including all the rear stairs, shelves, etc., with pure white 
lead and oil. 

All the woodwork in hasement and attic throughout, and bath rooms, and 
butler's pantry, to be grained in imitation of oak, and varnished twice over with 
the best varnish; also panelhng under wash basins; and all the rear wood cornices 
to be painted brown stone color and sanded. 

All the vestibule doors and trimmings to be gramed in imitation of dark oak, 
and varnished .... over. 

Glazing. — Glaze all the front windows throughout on all the stories, and lights to entrance 
doorway, and vestibule doors, with .... plate glass. 

Glaze all the windows of ... . with .... 

All the residue of the glazing to be done with . . . ., except under cellar. 

The outer skyhght to be glazed with extra double thick ribbed glass. 

Glaze all the sash doors throughout, and sashes -in partitions, with ornamented 
enamelled glass, to cost not less than .... per foot. 

All to be left whole and clean at the completion of the building. 

The skylight over the principal stairs to be glazed with stained glass to cost 
not less than %... 

The skylight over reception room to cost not less than $. ., and one over the pri- 
vate stairs to cost $ . . . 

Size of the glass to be obtained from the drawings. 

The owner to have the privilege of furnishing the stained glass, and the contrac- 
tor to deduct the amount specified. 

Finish and complete all the carpenters' work to the true meaning and intention of 
the drawings, and this specification ; leaving all the works sound, clean, and perfect 
at the completion of the building. All the floors and stairs are to be properly 
scrubbed before the last coat of paint is put on. 



Conditions which may he inserted in hoth Contracts, 
I 

The work is to be commenced on or before the . . day of . . . . , and the building 

to be completely finished and delivered to the owner by the 

The owner shall be held harmless, either in suits at law or otherwise, from all damages 
to the property of other person or persons arising during the progress of the work. And 
he shall not be responsible for any of the materials to be used in the erection of the build- 
ing that may be lost or stolen, or destroyed by fire, beyond the amount of any insurance 
that he may have on the building, until the last instalment on the contract is paid. Such 
insurance would be equally for the benefit of owner and contractor, pro rata. 



SHADING AND SHADOWS. 313 



SHADING Aj^D shadows. 

Light is diffused tlirongli space in straight lines, and the lines of light 
are called rays. When the source of light is situated at a very great dis- 
tance from the illuminated objects, as in the case of the sun -with relation 
to the earth, the rays of light do not sensibly diverge, and may be regarded 
as exactly parallel to each other. Such is the case in mechanical draw- 
ings, where the objects to be represented are always regarded as illumi- 
nated by the solar light. 

Light is called direct when it is transmitted to an object without the 
intervention of any opposing medium. But as all bodies subjected to the 
action of light possess, in a greater or less degree, the property of giving 
out a certain portion of it to the surrounding objects, this reflected light 
becomes in its turn, though with greatly diminished intensity, a source of 
illumination to those objects which are deprived of direct light. 

Everything which tends to intercept or prevent the direct light from 
falling in upon a body, produces upon the surface of that body a degree 
of obscurity of greater or less intensity ; this is called a shade or shadow. 
Such effects are usually classified as direct shadows and cast shadows. 

Tlie shade proper, or direct shadow^ is that which occurs on that por- 
tion of the surface of a body which is situated opposite to the enlightened 
part, and is the natural result of the form of the body itself, and of its posi- 
tion with regard to the rays of light. The cast shadow, on the other hand, 
is that which is produced upon the surface of one body by the interposi- 
tion of another between the former and the source of light ; thus inter- 
cepting the rays which would otherwise illuminate that surface. An 
illustration of this distinction is afforded in the pp*amid represented at 
fig. 1, Plate LXXX., where the shade proper is shown upon that half of 
the figure which is denoted by the letters D' E' G^ F' in the plan, while the 
cast shadow occupies the space comprised between the lines E' e and F^ d 
on the horizontal plane of projection. Cast shadows may also obviously 



314 SHADING AND SHADOWS. 

be produced upon the surface of a body by tlie form of the body itself ; as, 
for example, if it contain projecting or concave parts. 

Tlie limit of the direct shadow in any body, whatever may be its form 
or position, is a line of greater or less distinctness, termed the line of sep- 
aration hetween light and shade j or, more shortly, the line of shade j this 
line is, of course, determined by the contact of the luminous rays with the 
surface of the body ; and if these rays be prolonged till they meet a given 
surface, by joining all the points of intersection with that surface, we ob- 
tain the outline of the shadow cast upon it by the part of the body which 
is deprived of light. 

The rays of light being regarded as parallel to each other, it is obvious 
that in the delineation of shadows, it is only necessary to know the direc- 
tion of one of them ; and as that direction is arbitrary, we have adopted 
the usual and confessedly the most convenient mode of regarding the rays 
as in all cases falling in the direction of the diagonal of a cube, of which 
the sides are parallel to the planes of projection. Tlie diagonal in j)rojec- 
tion upon the vertical and horizontal planes lies at an angle of 45° with 
the ground-line ; and thus the light in both elevation and plan appears at 
the angle of 45°. In illustration, let E, P/ (fig. 1, pi. LXXIX.) be the pro- 
jections of a ray of light in elevation and plan ; and let A, A^, those of a 
point of which the shadows are required to be projected upon the vertical 
plane X Y. Draw the straight lines A a^ A! a\ parallel to the lines E, E^, 
and from a\ where the line A^ a' meets the plane X Y, draw the perpen- 
dicular a' a to meet the oblique line K a; then the intersection a is the 
position of the shadow of the point A. 

In the following illustrations, the same letter accented, is employed in 
the j)lan as in the elevation, to refer to the same point or object. 

The projections of the diagonals of the imaginary cube which denote 
the direction of the rays of light being equal in both planes, it follows 
that in all cases, and whatever may be the form of the smface upon which 
the shadow is cast, the oblique lines joining the projections of the point 
which throws the shadow, and that w^hich denotes it, are also equal. Thus 
the line A a in the elevation is equal to the line A^ a' in the plan. Hence 
it will in some cases be found more convenient to use the compasses 
instead of a geometrical construction ; as, for example, in place of project- 
ing the point a' by a perpendicular to the ground-line, in order to obtain 
the position of the required shadow «, that point may be found by simply 
setting off upon tlie line K a i\. distance equal to A' a' . 

Plate LXXIX., fig. 1. — Required to determine the shadow cast upon 
the vertical wall 'X.Y l>y the straight line A B. 



PLATE LXXIX, 



—tJsj 





^\ 










w- 




-^ 




'-- -\ \ 










% 


^ 









SHADING AND SHADOWS. 315 

It is obvious tliat in this case the shadow itself will he a straight line ; 
hence, to solve the problem, it is only necessary to find two j^oints in that 
line. We have seen that the position of the shadow thrown by the point 
A is at a; by a similar process we can easily determine the point h, the 
position of shadow thrown by the oj^posite extremity B of the given line ; 
the straight line a Z>, Avhich joins these two points, is the shadow required. 

It is evident from the construction of this figure, that the line a h is 
equal and parallel to the given line A B ; this results from the circum- 
stance that the latter is parallel to the vertical plane X Y. Hence, whe7i 
a line is i)arallel to a jplane^ its shadow ujpon that jplane is a line which is 
equal and jparallel to it. 

Suppose now that, instead of a mere line, a parallel slip of wood or 
paper, A B C D, be taken, which, for the sake of greater simj^licity, we 
shall conceive as having no thickness. The shadow cast by this object 
upon the same vertical plane X Y is a rectangle ah c d^ equal to that 
which represents the projection of the slip, because all the edges of the 
latter are parallel to the plane upon which the shadow is thrown. Hence, 
in general, when any surface., whatever may be its form, is jparallel to a 
"plane., its shadow thrown iijpon thatjplane is a figure similar to it., and simi- 
larly situated. This principle facilitates the delineation of shadows in 
many cases. In the present example, an idea may be formed of its 
utility ; for, after having determined the position of any one of the points 
«, 5, c, J, the figure may be completed by di'awing lines equal and parallel 
to the sides of the slip, without requiring to go through the operations in 
detail. 

Fig. 2. — When the object is not parallel to the given plane, the cast 
shadow is no longer a figure equal and similarly placed ; the method of 
determining it remains, however, unchanged ; thus, take the portion A E 
of the slip A B, which throws its shadow on the plane X Y ; draw the 
lines A (2, E (?, C (?, ry, and A' a' ^ YJ e\ 23arallel to the rays of light ; make 
A a and C c equal to A' a'\ and E e and F/* equal to E' e' ; connect a efc^ 
and we have the outline of the shadow of the slij) A E. 

By an exactly similar construction we have the shadow of the portion 
E B on the plane Y Z, which being inclined to the plane of projection in 
a direction contrary to X Y, necessarily causes the shadow to be broken, 
and the part e d to lie in a contrary direction to af. 

Fig. 3 still further illustrates the determination of the shadow of the 
slip upon a moulding placed on the plane X Y parallel to the slip. 

Fig. 4. — To find the shadoio cast hy a straight line A B n^07i a curved 
surface., either convex or concave, whose horizontal projection is repre- 
sented hy the line X e' Y. 



316 SHADING AND SHADOWS. 

It has been already explained, that the shadow of a point upon any 
surface whatever is found by drawing a straight line through that point, 
parallel to the direction of the light, and marking its intersection with the 
given surface. Therefore, through the projections A and A' of one of the 
points in the given straight line, draw the lines A a^ K! a\ at an angle of 
45°; and through the point a\ where the latter meets the projection of the 
given surface, raise a perpendicular to the ground-line ; its intersection 
with the line A (^ is the position of the shadow of the first point taken ; 
and so for all the reversing points in the line. 

If it be required to delineate the entire shadow cast by a slip A B C D, 
by the construction above explained, trace two equal and parallel curves 
aet, cfcl^ which will represent the shadows of the sides A B and C D ; 
while those of the remaining sides will be found denoted by the vertical 
straight lines a c and h d^ also equal and parallel to each other, and to the 
corresponding sides of the figure, seeing that these are themselves vertical 
and parallel to the given surfaces. 

rig. 6. — ^When the slip is placed j)erpendicularly to a given plane XY, 
on which a projecting moulding, of any form whatever, is situated, the 
shadow of the upper side A' B', w^hich is projected vertically in A, will be 
simply a line A (^ at an angle of 45°, traversing the entire surface of the 
moulding, and prolonged unbroken beyond it. Tliis may easily be de- 
monstrated by finding the position of the shadow of any number of points 
such as D^, taken at pleasure upon the straight line A' B^ Tlie shadow 
of the opposite side, projected in C, will follow the same rule, and be de^ 
noted by the line C c, parallel to the former. Hence, as a useful general 
rule : in all cases where a straight line is perpendicular to a plane ofpror 
jection^ it throws a shadow upon that plane in a straight line^ forming an 
angle of 4:5° with the ground-line. 

Fig. 6 represents still another example of the shadow cast by the slip 
in a new position ; here it is supposed to be set horizontally in reference 
to its own surface, and perpendicularly to the given plane X Y. Here 
the shadow commences from the side D B, which is in contact with this 
plane, and terminates in the horizontal line a c, which corresponds to the 
opposite side A C of the slip. 

Plate LXXIX., fig. 7. — Bequired to find the shadow cast upon a verti- 
cal plane "KY hj a given circle parallel to it. 

Let C, C, be the projections of the centre of the circle, and K, E', those 
of the rays of light. 

It lias been ah'cady shown, that when a figure is parallel to a plane, 
its shadow cast upon that plane is a figure in every respect equal to, and 



SHADING AXD Sn.\JDOWS. 317 

symmetrical witli it ; therefore tlie sliadow cast by tlic circle now under 
consideration will be expressed by another circle of equal radius ; conse- 
quently, if the j)Osition of the centre of this new circle be determined, the 
problem will be solved. Now the jDosition of the shadow of the central 
point C, according to the rules already fully developed, is easily fixed at 
c; from which point, if a circle equal to the given circle be described, it 
will rej)resent the outline of the required shadow. 

Fig. 8. — When the circle is perpendicular to both planes of projection, 
its projection upon each will obviously be represented by the equal diam- 
eters A B and C T)\ both perpendicular to the ground-line. In this case, 
to determine the cast shadow, describe the given circle upon both planes, 
as indicated by the figures, and divide the circumference of each into any 
number of equal parts ; then, having projected the points of division, as 
A^, C^, E'^, &c., to their respective diameters A B and C D', di-aw from 
them lines parallel to the rays of light, which, by their intersection with 
the given plane, will indicate so many points in the outline of the cast 
shadow. 

Fig. 11. — If the given circle be horizontal, its shadow cast upon the ver- 
tical plane X Y becomes an ellipse which must be constructed by means 
of points, as indicated by the figures referred to above ; that is to say, that 
in the circumference of the circle a certain number of points are to be 
taken, such as A' D' B^, &c., which are to be projected successively to 
A, D, B, on the line A B, and through each of these points lines are to be 
drawn parallel to the direction of the rays of light, and their intersection 
with the given plane determined. The junction of all these points will 
give the ellipse adh^ which is the contour of the required shadow. 

Fig. 9 represents a circle whose plane is situated perpendicularly to 
the direction of the luminous rays. In this example the method of con- 
structing the cast shadow does not difi'er from that pointed out in reference 
to fig. 11, provided that both projections are made use of. But it is obvious, 
that instead of laying down the entire horizontal projection of this circle, 
all that is necessary is to set ofi* the diameter D' E' equal to A B, because 
the shadow of this diameter, transferred in the usual way, gives the major 
axis of the ellipse which constitutes the outline of the shadow sought, 
while its minor axis is at once determined by a 5, equal and parallel to 
AB. 

Fig. 10 exhibits the case of a circle parallel to the vertical plane of pro- 
jection, throwing its shadow at once upon two plane surfaces inclined to 
each other. To delineate this shadow, all that it is necessary specially 
to point out is, that the points d and e are found by drawing from T a 



318 SHADING AND SHADOWS. 

line Y D', parallel to the rays of light, and projecting the point D' to D 
andE. 

Fig. 12 represents constructions similar to the foregoing, for obtaining 
the form of the shadow cast by a horizontal circle npon a vertical curved 
surface. 

We may here remark, that in every drawing where the shadows are to 
be inserted, it is of the utmost importance that the projections which re- 
present the object whose shadow is required should be exactly defined, aa 
well as the surface upon which this shadow is cast ; it is therefore advis- 
able, in order to prevent mistakes and to insure accuracy, to draw the 
figures in China ink, and to erase all pencil marks before proceeding to 
the operations necessary for finding the shadows. 

Plate LXXX., fig. 1. — To find the outline of the shadow cast upon hoth 
jplanes of projection hy a regidar hexagonal pyramid. 

In these figures it is at once obvious, that the three sides A^ B' F', 
A' B^ C^, and A^ C T>' alone receive the light ; consequently the edges 
A^ F' and A^ D^ are the lines of shade. To solve this problem, then, we 
have only to determine the shadow cast by these two lines, which is 
accomplished by drawing from the projections of the vertex of the pyra- 
mid the lines A V and A' a' parallel to the ray of light, then raising from 
the point V a perpendicular to the ground line, which gives at a' the 
shadow of the vertex on the horizontal plane, and finally by joining this 
last point a' with the points J)' and F^ ; the lines D^ a' and F^ a' are the 
outlines of the required shadow on the horizontal plane. But as the pyra- 
mid happens to be situated sufficiently near the vertical plane to throw a 
portion of its shadow towards the vertex upon it, this portion may be 
found by raising from the point c, where the line A^ a' cuts the ground-line, 
a perpendicular c <2, intersecting the line A h' in a; the lines a d and a e 
joining this point with those where the horizontal part of the shadow 
meets the ground-line, will be its outline upon the vertical plane. 

Fig. 2. — Beqidred to determine the limit of shade on a cylinder placed 
vertically^ and lihewise its shadow cast upon the two planes of projection. 

TJie lines of shade on a cylinder situated as indicated, are at once found 
by drawing two tangents to its base, parallel to the ray of light, and pro- 
jecting tlirough the points of contact lines parallel to the axis of the 
cylinder. 

Draw tlie tangents D' d' and C^ c' parallel to the ray E' ; these are the 
outlines of tlic shadow cast upon the horizontal plane. Through the point 
of contact C draw the vertical line Q>' Q' ; this line denotes the line of shade 
upon the surface of the cylinder. It is obviously unnecessary to draw the 



PLATE LXXX. 



318 



Fig. 1. 



I'i-. 2. 




Fig. 3. 



Fi- 4. 



|!'':;:::;iii:ii,ii;iiiiiii;iiiiiyiy;::!;iiii!;n 



J B 



R^ 



m It^Mi 



^m ii 



'^ 



iliii 






l!""l!lll'!!!"l!!!l!!!!! 






— yrt: — r /nfiMirinfifiiiiriiiTiTiririiiifii ifiriiiiiiii'.' 




71/ i/' 



^m^ 



•3 l'-::ll'!!!!"'!':!!!!!-^!^nnnnnr??>^ -vr. 



/?, 



■■■'A'- 








F'-- 




JB r 




1^ 




l'l!ll:li'li|ii!J 






K f 











i 





AJi D ^ c m 



^mi 



SHADING AND SHADOWS. 319 

perpendicular from tlie opposite point D', because it is altogether con- 
cealed in the vertical elevation of the solid. In order to ascertain the 
points C and D' with accuracy, draw through the centre O' a diameter 
perpendicular to the ray of light R^ 

Had this cylinder been placed at a somewhat greater distance from the 
vertical plane of projection, its shadow would have been entirely cast upon 
the horizontal j^lane, in wdiich case it would have terminated in a semi- 
circle drawn from the centre o\ with a radius equal to that of the base. 
But as a portion of the shadow of the upper part is thrown upon the ver- 
tical plane, its outline will be defined by an ellipse drawn in the manner 
indicated in fig. 11, plate LXXIX. 

Fig. 3. — To find the line of shade in a reversed cone, and its shadow 
cast ujpon the tioo planes of ^projection. 

From the centre A^ of the base draw a line parallel to the ray of light ; 
from the point a' where it intersects the perpendicular, describe a circle 
equal to the base, and from the point A^ draw the lines A' V and A^ c\ touch- 
ing this circle ; these are the outlines of the shadow cast upon the horizon- 
tal plane. Then from the centre A' draw the radii A' B^ and A' Q>' parallel 
to a' V and a' c' ; these radii are the horizontal projections of the lines of 
shade, the former of wdiich, transferred to B D, is alone visible in the ele- 
vation. But in order to trace the outline of that portion of the shadow 
which is thrown upon the vertical plane, it is necessary to project the 
point C to C, from which, by a construction which will be manifest from 
inspection of the figures, we derive the point c and the line c d as part of 
the cast shadow of the line C^ A^ The rest of the outline of the vertical 
portion of the cast shadow is derived from the circumference of the base, 
as in fig. 2. 

Fig. 4. — When the cylinder is placed horizontally, and at the same 
time at an angle with the vertical plane, the construction is the same as 
that explained (fig. 2) ; namely, lines are to be drawn parallel to the ray 
of light, and touching the opposite points of either base of the cylinder, 
and through the points of contact A and C the horizontal lines A E and 
C D are to be drawn, denoting the limits of the shade on the figure. The 
latter of "these lines only is visible in the elevation, while, on the other 
hand, the former, A E alone, is seen in the plan, where it may be found 
by drawing a perpendicular from A meeting the base F^ G' in A^ The 
line A^ E' drawn parallel to the axis of the cylinder is the line of shade 
required. Project the shadow of the line A E on the vertical plane as in 
previous examples, and the construction w^ill define the outline of the 
shadow of the cylinder. 



320 SHADma and shadows. 

The example liere given presents the j)articular case in which the base 
of the cylinder is parallel to the direction of the rays of light in the hori- 
zontal projection. In this case, all that is required in order to determine 
the line A^ E' is to ascertain the angle wliich the ray of light makes with 
the projection of the figure. Draw a tangent to the circle F^ A^ G' (which 
represents the base of the cylinder laid down on the horizontal plane), in 
such a manner as to make with F^ G^ an angle of 35° 16^, and through the 
point of contact A^ draw a line parallel to the axis of the cylinder ; this 
line E^ A^ will be the line of shade as before. 

Fig. 5 represents a cylinder upon which a shadow is thrown by a rec- 
tangular prism, of which the sides are parallel to the planes of projection. 
The shadow in this case is derived from the edges A' D' and A' E', the 
first of which, being perpendicular to the plane of projection, gives, accord- 
ing to principles already laid down, a straight line at an angle of 45° for 
the outline of its shadow, whereas the side A' E' being parallel to that 
plane, its shadow is determined by a portion of a circle ah c^ described 
from the centre o. 

Figs. 6, 7. — If the prism be hexagonal, or a cylinder be substituted for 
it, the mode of construction remains the same. But it should be observed, 
that it is best in all such cases to commence by finding the points which 
indicate the main direction of the outline. To ascertain the point a at 
which the shadow commences, draw from a' the line a' K! at an angle of 
45°, which is then to be projected vertically to a A. Tlien the highest 
j)oint h (fig. 7) should be determined by the intersection of the radius 
O^ B^ (drawn parallel to the ray), with the circumference of the base of 
the cylinder on which the required shadow is cast ; and finally, the point 
c, where the outline of the cast shadow intersects the line of shade, should 
be determined by a similar process. 

Fig. 1 represents a hexagonal prism upon which a shadow is thrown 
by a rectangular prism. Determine the point <^ as in fig. 5, pi. LXXX. ; 
draw from the angular points V ^ c' lines parallel to the direction of the 
light, intersecting the edge of the rectangular prism at B', Q' ; project these 
points, and draw the lines B Z*, C c; their intersections with the edges of 
the hexagonal prism will be the limiting points J, c, of the required 
shadow. 

Fig. 2 represents a hexagonal prism upon which a shadow is cast by 
another hexagonal prism. The construction is precisely similar to the 
preceding. Lines parallel to the direction of the light are drawn from 
the angular ])()iiits of botli interior and exterior prisms; these points are 
projected, and the limiting points «, J, c, <^, of the shadow arc determined. 



SHADING AND SHADOWS. 



321 



Fio-. 3 represents a hexagonal prism upon which a shadow is cast by a 
cylinder, a variety of the preceding; but as in this case the outline 
of the shadow is curved, in addition to the lines from the angular points 






ABTJ) 'H 



a en 



3 



Fig. 1. 



Fig. 2. 



Fig. 



of the prism, parallels are also drawn from as many intermediate points 
h' d\ as may be necessary to determine the outline of the curved shadow. 

Plate LXXXI., fig. 7. — To define the shadows cast iijpon the interior of 
a hollow cylinder in section hy itself and hy a circular piston fitted into it. 

The example shows a steam cylinder, A, in section, by a plane passing 
through its axis, with its piston and rod in full. 

Conceive, in the first instance, the piston P to be removed ; the shadow 
cast into the interior of the cylinder wdll then consist, obviously, of that 
projected by the vertical edge B C, and by a portion of the horizontal 
edge B A. To find the first, draw through B' a line B' I' at an angle of 
45° with B^ A ; the point h\ where this line meets the interior surface of 
the cylinder, being projected upwards to fig. 7, gives the line &/as the 
outline of the shadow sought. Then, parallel to the direction of the light, 
draw a tangent at F' to the inner circle of the base ; its point of contact 
being projected to F in the elevation, marks the commencement of the out- 
line of the shadow cast by the upper edge of the cylinder. The point 5, 
where it terminates, will obviously be the intersection of the straight line 
/ h already determined, with a ray B 1) from the upper extremity of the 
edge B C ; and any intermediate point in the curve, as <?, may be found by 
taking a point E^, between B' and h' ^ projecting it to E, and causing rays 
E ^, E^ e\ to pass through these points. The outline of the shadow required 
will then be the curve Y eh and the straight line hf Supj)ose now the 
piston P and its rod T to be inserted into the cylinder as shown. Tlie 
lower surface of the piston will then cast a shadow upon the interior sur- 
face of the cylinder, of which the outli.ie J) d h o^ may be formed in the 
21 



322 



SHADING- AND SHADOWS. 



same way, as will be obvious from inspection of tbe figm^es, and compari- 
son of the letters of reference. The piston-rod T being cylindrical and 
vertical, it casts also its shadow into the interior of the cylinder ; it will 
obviously consist of a rectangle ijlTc drawn parallel to the axis, and of 
which the sides ij and h I are determined by the tangents I' i' and K^ Ic' . 

Figure 4. — ^This example consists of a hollow cylinder, surmounted 
by a circular disc or cover, sectioned through the centre, where it is also 
penetrated by a cylindrical aperture. The construction necessary for find- 
ing the outlines of the cast shadow will obviously be the same as abeady 
laid down. In this case, however, it is proper to know beforehand what 
parts of the upper and lower edges of the central aperture cast their 
shadows into the interior of the cylinder ; if, then, we take the trouble to 
construct the shadows of each of these edges separately, we shall find that 
that of the upper edge is a curve t ef^ and that of the lower a similar 
curve a c e, cutting the former in c. Tliis point limits the parts of each 
curve which are actually visible ; namely, the portion 5 c of the first, and 
the portion e c oi the second ; hence it follows, that in order to avoid un- 
necessary work, we should first determine the position of the point of inter- 
section, c, of the two curves, which is in fact the cast shadow of the lowest 
point C in the curve D C, previously laid down in the circular opening of 
the cover, in the manner indicated in fig. T, plate LXXXI. 





Fig/ 5. 



Fig. 6.-Let a cylinder in section to be set at an inclination to the horizon- 
tal plane. To find the outline of the shadow cast into its interior, describe 
upon the prolongation of the axis of the cylinder a semicircle A' a' B', 
representing its interior surface, and tlien, in any convenient part of the 
paper, draw the diagonal on o parallel to the hue of light A' E', and con- 



i 



SHADING AND SHADOWS. 



32J 




Fig. 6, 



struct a square m n oxj (fig. C) ; from one of the extremities, o^ draw tlic 
line o r i^arallel to A' B', and tlirougli the opposite extremity, ?/2, draw a 
perpendicular r s to this line, and set off on the perpendicular the distance 
'/' s equal to the side of the square, and join s o. l^ow, 
draw tlirough the point A^, in the original figure, a line 
A' a\ parallel to s o, intersecting the circle A' a' B' in the 
point «', which being projected by a line parallel to the 
axis of the cylinder, and meeting the line A a, drawn at 
an angle of 45°, gives the first point a in the curve C d a. 
The other points will be obtained in like manner, by draw- 
ing at pleasure other lines, such as T)' d' ^ parallel to A' a'. 

To find the outline of the shadoio cast into the interior of a hollow 
hemisjphere. 

Let A B C D (fig. 7) represent the horizontal projection of a concave 
hemisphere. Here it is sufficiently 
obvious, that if we draw through 
the centre of the sphere a line per- 
pendicular to the ray of light A C, 
the points B and D will at once 
give the extremities of the curve 
sought. Construct the square (fig. 
6), making m o the diagonal paral- 
lel with A C at m; erect the per- 
pendicular on m^^ making ni m^ 
equal to one side of the square ; 
connect m^ o. Take now upon the 
prolongation of the line B D any 
point O', from which, as a centre, describe a semicircle with the radius 
A O, and from the point A' draw the straight line A' a parallel to om^ ; 
the point a' of its intersection with the circle A' a' Q\ projected to a^ will 
be another principal point in the outline of the shadow. 

By imagining similar sections, such as E F parallel to the former A C, 
and laying down in the same way semicircles E F corresponding to them, 
and drawing through E^ lines parallel to o m^, and projecting their intersec- 
tion with their semicircles to the corresponding sections E F on the plan, the 
remaining points in the curve sought may be obtained. But as this cmwe 
is an ellipse, of which the diameter D B is the major axis, and the line O a 
the half minor axis, it follows that this last line being determined, the 
curve may be constructed by the ordinary methods for ellipses. 




Fig. 7. 



321- 



SHADING AND SHADOWS. 



There will now be no difficulty in constructing the cast shadow in the 
interior of a concave surface (fig. 8), formed by the combination of a hollow 
semi-cylinder and a quadrant of a hollow sphere, called a niche, as we 
know the mode of tracing the shadows upon each of these figures separ- 
ately. Thus, the shadow of the circular outline upon the spherical por- 
tion is part of an ellipse 
icD, whose semi-axis major 
is O D ; the semi-axis minor 
is obtained by describing the 
semicircle B^ ^' E with the 
radius O B, drawing from 
the point B^ the straight line 



,*■ 



\P' 







>E 



B^ i' parallel to a line o m^, 
found by a construction as before (fig. 6), 
and finally projecting the point of intersec- 
tion i^ to i on the straight line B O. The 
point e, where this ellipse cuts the horizontal 
diameter A F, limits the cast shadow upon 
the spherical surface ; therefore all the points 
beneath it must be determined upon the cy- 
lindrical part. Through A^ in the plan draw 
the line A^ a' parallel to the ray of light ; 
project a^ till it intersects the line of light 
A a in the elevation at a. The line of shadow 
below a is the shadow of the edge of the cy- 
linder, and must therefore be a straight line. 
The line of shadow between a and e is pro- 
duced by the outline of the circular part fall- 
ing on a cylindrical surface, and is established as in previous constructions, 
by drawing lines parallel to the rays of light through different points, as 
B in the curved outline, and similar lines through the corresponding 
points B' in the plan, and projecting their intersections h' with the semi- 
circle till they intersect the first line at h as points in the line of shadow. 

Plate LXXXI., figs. 1, 2. — To find the line of shade in a sphere, and 
the outline of its shadow cast upon the horizontal plane. 

Tlie line of shade in a sphere is simply the circumference of a great 
circle of wliich the plane is perpendicular to the direction of the luminous 
ray, and consequently inclined to the two planes of projection. This line 
will, therefore, be represented in elevation and plan by two equal ellipses, 



Fig. 8. 



325 



PLATE LXXXI. 




kl-. 






SHADING AND SHADOWS. 325 

the major axes of wliicli are obviously the diameters C D and C D', drawn 
at an angle of 45°. 

To find the minor axes of these curves, assume any point O', upon the 
prolongation of the diameter of the perpendicular C D' (fig. 2), draw 
through this point the straight line O' o\ inclined at an angle of 35° 16', 
to A' B' or its parallel, and erect upon it the perpendicular E* F'. The 
projection of the two extremities E^ and F^ upon the line A^ B', will give, 
in the plan, the line E' F' for the length of the required minor axis of the 
ellipse, i. e. of the line of shade in the plan ; and this line being again 
transferred to the elevation, determines the minor axis E F of the line of 
shade in the elevation. 

Supposing it were required to draw these ellipses, not by means of 
their axes, but by points, any number of these may be obtained by making 
horizontal sections of the sphere. Thus, for example, if we draw the chord 
G H, parallel to A^ B', to represent one of these sections, and from the 
point a, where it cuts the diameter E" F', if we draw a j)erpendicular to 
A^ B', the points a' a\ where it intersects the circumference of a circle 
representing the section G H in plan, will be two points in the line of 
shade required. Tliese points may be transferred to the elevation, by 
supposing a section g h to be made in fig. 1, corresponding to G H in fig. 
2, and projecting the points a' a' by perpendiculars to g A, the line repre- 
senting the cutting plane. 

The outline of the shadow cast by the sphere upon the horizontal plane 
is also obviously an ellipse ; it may be constructed either by means of its 
two axes, or by the help of points, in the manner indicated in the figure. 

Figs. 3, 4, and 5. — To draio the line of shcule on the surface of a ring 
of circular section^ in vertical section^ elevation^ and plan. 

AYe shall first point out the mode of obtaining those primary points in 
the curve which are most easily found, and then proceed to the general 
case of determining any point whatever. 

If tangents be di-awn to the circles represented in figs. 3 and 4, parallel 
to the ray of light, their points of contact, «, &, <?, cZ, will be the starting 
points of the required lines of shade. 

Again, the intersections of the horizontal lines a <?, d g, c f drawn 
through these points, with the axis of the ring, will give so many new 
points <?, g^f in the curve. These points are denoted in the plan (fig. 5), by 
setting off the distances a e and c f upon the vertical line g' D, on both 
sides of the centre C. 

Farther, the diameter F' G', drawn at an angle of 45°, determines, by 
its intersections with the exterior and interior circumferences of the rino^, 



326 SHADING AND SHADOWS. 

four other points F^, t' ^ x\ and G^, in the curve in question ; these points 
are all to be projected vertically upon the line A B. 

And, lastly, to obtain the lowest points I Z, construct the two squares 
(fig. 6), making the diagonals o' m' and o m parallel severally to the ray 
of light in plan and elevation ; revolve o' in upon the point o' until it be- 
comes parallel with the vertical plane of projection; project to m'^, and 
connect o m^ ; draw tangents to the circles represented in figs. 3 and 4, 
parallel to the line o m^, and transfer the distances between the points of 
contact, <§, <9, and the axis of the ring, to the radius E^ C (fig. 5), where 
they are denoted by V V / these latter points are then to be projected ver- 
tically to Z, Z, upon the horizontal lines drawn through the same points s^ s 
(figs. 3 and 4). 

The curves sought might now, in most instances, be traced by the 
points thus obtained ; but should the ring be on a large scale, and great 
accuracy be required, it may be proper to determine a greater number 
of points. For this purpose, draw through the centre C\ a straight line 
V H^, in any direction, draw through o', one of the angular points of the 
cube at fig. 6, a straight line or parallel to V H^, and from the opposite 
point m' draw a perpendicular m' / to o' r' . Then, having revolved the 
point t' to T^ by means of a circular arc, in order to admit of this last 
point being projected to r, we join o r. 

Applying this construction to the figures before us, we now draw tan- 
gents to the circles represented in figs. 3 and 4, parallel to the line o r^ 
and, taking as radii the distances from their respective points of contact, 
h and I, to the axis of the ring, we describe corresponding circles about 
the centre C, fig. 4. We thus obtain four other points in the curves re- 
quired, namely, F, i, A, and H, which may also be projected upon the 
horizontal lines drawn through the points h or I. 

By drawing the straight line J^ K^ so as to form with F^ G-^ the same 
angle which the latter makes with the line H^ I^, we obtain, by the inter- 
section of that line with the circles last named, four other points of the 
curves in question. 

Figs. 8 and 9. — Of the shadows cast lopon the surfaces of grooved pulleys. 

The construction of cast shadows upon surfaces of the kind now under 
consideration is founded upon the principle already announced, that when 
a circle is jparallel to aplane^ its shadow^ cast iipon that plane ^ is another 
circle equal to the original circle. 

Take, in the first place, the case of a circular-gi'ooved pulley (figs. 8 
and 9) ; the cast shadow on its surface is obviously derived from the cir- 
cumference of the upper edge A B. To detcrinine its outline, take any 



32T 



PLATE LXXXII. 



r JilJ'!!:__o'_ 




I I'kiiiiiiift'JW^^^ 




Fig. 1. 



r/ 



/ I 



t -.< 



\A' 



Fig. 2. 




Y "-G^ 



Fig. 5. 

'7o 



i 



y/V / / 



Fig. S, 



Fig. 6. 



#^^«^^ 



"""""I'llllll, 







i,u*iiil 









jAA..A.± 



4t 



rig. 4. 










SHADING AND SHADOWS. 827 

horizontal line D E upon fig. 8, and describe from the centre C (fig. 9) a 
circle with a radius equal to tlie half of that line ; then draw, through the 
same centre, a line parallel to the raj of light, wliich will intersect the 
plane T> E in c ; lastly, describe from the point c\ as a centre, an arc of a 
circle with a radius equal to A C ; the point of intersection, a\ of this arc, 
with the circumference of the plane D E, will give when projected to a 
(fig. 8), one of the points in the curve required. 

To avoid unnecessary labor in drawing more lines parallel to D E than 
are required, it is important, in the first place, to ascertain the highest 
point in the curve sought. This point is the shadow of that marked II on 
the upper edge of the pulley, and which is determined by the intersection 
of the ray C W with the circumference of that edge in the plan ; and it 
is obtained by drawing through the point A (fig. 8) a straight line at an 
angle of 35° 16^ with the line A B, and through the point e, striking a 
horizontal line e /*, which by its intersection with the line H 7^, drawm at 
an angle of 45°, will give the point sought. 

In fig. 9, the pulley is supposed to be divided horizontally in the centre, 
and the sliadow represented is derived from the smaller circle I K, and is 
easily constructed by methods above described. 

Plate LXXXII. — To trace the outlines of the shadows cast upon the 
surfaces of screws and nuts, hoth triangidar and square-threaded. 

Figs. 1 and 2 represent the projections of a screw wdth a single square 
thread, and placed in a horizontal position. A' a' being the direction of the 
ray of light. In this example, the shadow to be determined is simply that 
cast by the outer edge, A B, of the thread iq^on the surface of the inner 
cylinder ; therefore its outline is to be delineated in the same manner as we 
have already pointed out, in treating of a cylinder surmounting another 
of smaller diameter (page 320). 

Figs. 3 and 4. — ^The case of a triangular-threaded screw does not admit 
of so easy a solution as the above, because the outer edge A C D of the 
thread, in place of throwing its shadow upon a cylinder, projects it upon a 
helical surface inclining to the left, of which the generatrix is known. De- 
scribe from the centre O (fig. 3) a number of circles, representing the 
bases of so many cylinders, on the surfaces of which we must suppose 
helical lines to be traced, of the same pitch with those which form the ex- 
terior edges of the screw (see fig. 4). We must now draw any line, such 
as B' E', parallel to the ray of light, and cutting all the circles described 
in ^g. 3 in the points B^, F^, G', E^, which are then to be successively pro- 
jected to their corresponding helical lines in fig. 4, where they are denoted 
by B^, F, G, and E. Then, transferring the point B' (fig. 3) to its appro- 



328 SHADING AND SHADOWS. 

priate position B on the edge A C D (fig. 4), and drawing througli the 
latter a line B & at an angle of 45°, its intersection with the curve B^ G E 
will give one point in the curve of the shadow required. In the same 
manner, by constructing other curves, such as H^ J K, the remaining 
points, as 7i, in the curve may be found. 

Figs. 5 and 6. — ^The same processes are requisite in order to determine 
the outlines of the shadows cast into the interior surfaces of the nut cor- 
responding to the screw last described, as will be evident from inspection 
of figs. 5 and 6. These shadows are derived not only from the helical 
edge A B D, but also from that of the generatrix A C. 

Figs. Y and 8. — ^The shadow cast by the helix ABC upon the concave 
surface of the square-threaded nut is a curve ah Q, which is to be deter- 
mined in the same way as that in the interior of a hollow cylinder. The 
same observation applies to the edges A A*^ and A^ E, as well as to those 
of the helix F Gr H and the edge H I. With regard to the shadow of the 
two edges J K and K L, they must obviously follow the rules laid down 
in reference to figs. 4 and 6, seeing that it is thrown upon an inclined heli- 
cal surface, of which A L is the generatrix. 

The principles so fully laid down and illustrated in the preceding pages 
will be found to admit of a ready and simple application to the delineation 
of the shadows of all the ordinary forms and combinations of machinery 
and architecture, however varied or complicated ; and the student should 
exercise himself, at this stage of his progress, in tracing, according to the 
methods above explained, the outlines of the cast shadows of pulleys, spm*- 
wheels, and such simple and elementary pieces of machinery. It miist be 
observed, that the student should never coj)y the figures as here repre- 
sented, but should adopt some convenient scale somewhat larger than our 
figures, and construct his drawings according to the description, looking to 
the figures as mere illustrations ; in this way the principles of the con- 
struction will be more surely understood, and more firmly fixed in his 
mind. 

MANIPULATION OF SHADING AND SHADOWS, METHODS OF TINTING. 

The intensity of a shade or shadow is regulated by the various peculi- 
arities in the forms of bodies, and by the position which objects may 
occupy in reference to the light. 

Surfaces in the light. — Flat surfaces wholly exposed to the light, and 
at all points equidistant from the eye, should receive a uniform tint. 

In geometrical drawings, where the visual rays are imagined parallel 
to the plane of projection, every surface parallel to this plane is supposed 



SHADING AND Sn^UX)W5. 329 

to liave all its parts at the same distance from tlie eye ; siieli is the vertical 
side of the prism ah c d (fig. 4, plate LX XXIII). 

When two surfaces thus situated are parallel, the one nearer the eye 
should receive a lighter tint than the other. Every surface exposed to the 
light, but not parallel to the j)lane of projection, and therefore having no 
two points equally distant from the eye, should receive an unequal tint. 
In conformity, then, with the preceding rule, the tint should gradually in- 
crease in depth as the j)arts of such a surface recede from the eye. This 
eflect is represented in the same figure on the surface, a dfe^ which, by 
reference to the plan (fig. 1), is found to be in an inclined position. 

If two surfaces are unequally exposed to the light, the one which is 
more directly opposed to its rays should receive the fainter tint. 

Tims the face e' a' (fig. 1), presenting itself more directly to the rays of 
light than the face a' V^ receives a tint which, although graduated in con- 
sequence of the inclination of this face to the plane of projection, becomes 
at that part of the surface situated nearest to the eye fainter than the tint 
on the surface a h. 

Surfaces in shade. — TVIien a sm-face entirely in the shade is parallel to 
the plane of projection, it should receive a nniform dark tint. 

"When two objects parallel to each other are in the shade, the one 
nearer the eye should receive tlie darker tint. 

When a surface in the shade is inclined to the plane of projection, those 
parts which are nearest to the eye should receive the deepest tint. 

The face hg he (fig. 4), projected horizontally at h' g' (fig. 1), is situated 
in this manner. It will there be seen, that towards the line h c the tint is 
much darker than it is ^liere it approaches the line g h. 

If two surfaces exposed to the light, but unequally inclined to its rays, 
have a shadow cast upon them, that part of it which falls upon the smiace 
more directly influenced by the light should be darker than where it falls 
upon the other surface. 

Exemplifications of the foregoing rules may be seen on various figm^es 
in the plates. 

In order that these rules may be practised with proper efiPect, we shah 
give some directions for using the brush or hair-pencil, and explain the 
usual methods employed for tinting and shading. 

The methods of shading most generally adopted are either l)y the 
superposition of any number of flat tints, or by tints softened off" at their 
edges. Tlie former method is the more simple of the two, and should be 
the first attempted. 

Shading hy flat tints, — Let it be proposed to shade the prism (fig. 4, 



330 SHADma and shadows. 

plate LXXXIII.), bj means of flat tints. According to the position of the 
prism, as shown by its plan (fig. 1), the face alcd (fig. 4) is parallel to the 
plane of projection, and therefore entirely in the light. This face should 
receive a uniform tint either of India ink or sepia. When the surface to be 
tinted happens to be very large, it is advisable to put on a very light tint 
first, and then to go over the surface a second time with a tint sufficiently 
dark to give the desired tone to the surface. 

The face hgJiG being inclined to the plane of projection, as is shown 
by the line V g' in the plan (fig. 1), should receive a graduated tint from 
the line h c to the line g h. This graduality is obtained by laying on a 
succession of flat tints in the following manner : — First, divide the line 
1)' g' (fig. 1) into equal parts at the points 1^, 2^, and from these points pro- 
ject lines upon, and parallel to, the sides of the face hg he (fig. 4). These 
lines should be drawn very lightly in pencil, as they merely serve to cir- 
cumscribe the tints. A greyish tint is then spread over that portion of the 
face hgJic (fig. 2), between the lines h c and 1, 1. When this is dry, a 
similar tint is to be laid on, extending over the space comprised within the 
lines 1) G and 2, 2 (fig. 3). Lastly, a third tint covering the whole surface 
I cJig (fig. 4) imparts the desired graduated shade to that side of the prism. 
Tlie number of tints designed to express such a graduated shade depends 
upon the size of the surface to be shaded ; and the depth of tint must vary 
according to this number. 

As the number of these washes is increased, the whole shade gradually 
presents a softer appearance, and the lines which border the difl'erent tints 
become less harsh and perceptible. For this reason the foregoing method 
of representing a shade or graduated tint by washes successively passing 
over each other is preferable to that sometimes employed, of first covering 
the whole surface 1) g h c Avith a faint tint, then putting on a second tint 
h 2 2 c, followed, lastly, by a narrow wash h 1 1 c; because, in following 
this process, the outline of each wash remains untouched, and j^resents, un- 
avoidably, a prominence and harshness which, by the former method, are 
in a great measure subdued. 

The face a dfe is also inclined to the plane of projection, as shown by 
the lino a' e' in the plan (fig. 1) ; but as it is entirely in the light, it should 
be covered by a series of much fainter tints than the surface 1) g li <?, which 
is in the shade, darkening, however, towards the line ef. The gradation 
of tint is efi'cctcd in tlic same way as on the fiice 1) g li c. 

Let it be proposed to shade a cylinder (fig. 12), by means of flat tints : 

Li shading a cyliiulcr, it will be necessary to consider the difference in 
the tone proper to be maintained between the part in the light and that in 



PL.LXXXIII. 



'«(). 




J A 



Fig.Z. 



C I 2 



b I 




Ff\q 12. 



Fi^./I. 



Fl^. 10. 



Fig.^. 



Fig. 8. 



Fig. 7. 



Fig. 6. 




9 I 



liii 



m--^ 



b d. 



A 






^ 



y. - J; i li\i\X1>„ 



T ' V 




SHADING AND SHADOWS. 661 

tlie shade. It should be remembered that the line of separation between 
the light and shade ab is determined by the radius O a' (fig. 5), drawn 
perpendicular to the rays of light E. O. That part, therefore, of the cylin- 
der which is in the shade is comprised between the lines a h and c d. Tliis 
portion, then, should be shaded conformably to the rule previously laid 
down for treating surfaces in the shade inclined to the plane of projection. 
All the remaining part of the cylinder which is visible presents itself to 
the light ; but, in consequence of its circular figure, the rays of light form 
angles varying at every part of its surface, and consequently this surface 
should receive a graduated tint. In order to represent with effect the 
rotundity, it will be necessary to determine with precision the part of the 
surface which is most directly affected by the light. Tliis part, then, is 
situated about the line e i (fig. 12), in the vertical plane of the ray of light 
K O (fig. 5). As the visual rays, however, are perpendicular to the ver- 
tical plane, and therefore parallel to Y O, it follows that the part which 
appears clearest to the eye will be near this line Y O, and may be limited 
by the line T O, which bisects the angle Y O E and the line E O. By 
projecting the points e' and m', and drawing the lines e i and m n (fig. 12), 
the surface comprised between these lines will represent the lightest part 
of the cylinder. 

Tliis part should have no tint upon it whatever if the cylinder happen 
to be polished ; a turned iron shaft or a marble column for instance ; but 
if the surface of the cylinder be rough, as in the case of a cast-iron pipe, 
then a very light tint — considerably lighter than on any other part — ^may 
be given it. 

Again, let us suppose the half-plan of the cylinder y^ in' a' c' (fig. 5), to 
be divided into any number of equal parts. Indicate these divisions upon 
the surface of the cylinder by faint pencil lines, and begin the shading by 
laying a tint over all that part of the cylinder in the shade a c dh (fig. 6). 
riiis will at once render evident the light and dark parts of the cylinder. 
When this is dry, put on a second tint covering the line a h of separation 
of light and shade, and extending over one division, as shown in fig. 7. 
A third tint should be spread over this division, and one on each side of 
it, as in fig. 8. Proceed in this Avay until the whole of that part of the 
cylinder which is in the shade is coA^ered. The successive stages of this 
process may be seen in figs. 9, 10, and 11. 

Treat in a similar manner the '^ai't f ei g^ and complete the operation 
by covering the whole surface of the cylinder — excepting only the division 
e m n i (fig. 12) — with a very light tint ; the cylinder will then assume the 
appearance presented by fig. 12. 



332 SHADING AND SHADOWS. 

Shading hy softened tints. — ^The great advantage which this method 
possesses over the one just described, consists in imparting to the shade a 
much softer appearance ; the limitations of the different tints being imper- 
ceptible. On the other hand, it is considerably more difficult, requiring 
longer practice, and greater mastery over the movements of the brush to 
accomplish it with tolerable precision. 

Let it be proposed to shade by this method the segment of the hexa- 
gonal pyramid (fig. 8, plate LXXXIY.) 

The plan of this figure is similar to that of the prism (fig. 4, plate 
LXXXIII.) Its position in reference to the light is also the same. Thus 
the face ahcd should receive a uniform flat tint. If, however, it be de- 
sired to adhere rigorously to the preceding rules, the tint may be slightly 
deepened as it approaches the top of the pyramid, seeing that the surface 
is not quite parallel to the vertical plane. 

The face hg ho being inclined and in the shade, should receive a dark 
tint. The darkest part of this tint is where it meets the line h c, and grad- 
ually becomes lighter as it approaches the line g 7i. To produce this 
efifect, apply a narrow strip of tint to the side h c (fig. 6), and then, quali- 
fying the tint in the brush with a little water, join another strip to this, 
and finally, by means of another brush moistened with water, soften off this 
second strip towards the line 1, 1, which may be taken as the limit of the 
first tint. This is shown in fig. 6. 

When the first tint is dry, ^over it with a second, which must be simi- 
larly treated, and should extend beyond the first up to the line 2, 2 (fig. 
7). Proceed in this manner with other tints, until the whole face hg ho 
is shaded, as presented in fig. 8. 

In the same way the face e a df is to be covered, though with a con- 
siderably lighter tint, for the rays of light hajjpen to fall upon it almost 
perpendicularly. 

It may be observed, that consistently to carry out the rules we have 
laid down, the tint on these two faces should be slightly graduated from 
e a iofd^ and from c hioh g. But this exactitude may be disregarded 
until some proficiency in shading has been acquired. 

It is now proposed to shade the cylinder (fig. 4) by means of softened 
tints. The boundary of each tint being indicated in a manner precisely 
similar to that shown by fig. 5, plate LXXXIII., the first strip of tint must 
cover the line of extreme shade ah, and then be softened off on each side, 
as shown in fig. 13. Other and successively wider strips of tint are to fol- 
low, and receive the same treatment as the one first put on. The results 
of this process are shown in figs. 2, 3, and 4. 



LXXXIV. 



y: 



TfV 



%> "^. 



Fi^. J. 



a '^ 0/ L' 



9^ 



/- 




Fig.Z. 



a c 




Fi^.t. 



f^ 



Ml Ml 



a- c 

I 




Fiff.5. 




SUADINQ AND SUAD0W3. 333 

As this method requires considerable practice before it can be per- 
formed with much nicety, the learner need not be discouraged at the fail- 
ure of his first attempts, but persevere in practising on simple figures of 
diflferent sizes. 

If, after shading a figure by the foregoing method, any very apparent 
inequalities present themselves in the shade, such defects may be remedied 
in some measure by washing oflT redundancies of tint with a brush or a 
damp sponge, and by supplying a little color to those parts which are too 
light. 

Dexterity in shading figures by softened tints will be facilitated in prac- 
tising upon large surfaces ; this will be the surest way of overcoming that 
timidity and hesitation which usually accompany all first attemj)ts, but 
which must be laid aside before much proficiency in shading can be ac- 
quired. 

ELxiEOEATIOK OF SHADING AND SHADOWS. 

Thus far the simplest primary rules for shading isolated objects have 
been laid down, and the easiest methods of carrying them into o^^eration 
explained. It is now proposed to exemplify these rules upon more com- 
plex forms, to show where the shading may be modified or exaggerated, 
to introduce additional rules more especially adapted for mechanical color- 
ing, and to offer some observations and directions for efi'ectively shading 
the drawing of machines in their entirety. 

TVhatman's best rough-grained drawing-paper is better adapted for 
receiving color than any other. Of this paj)er, the double elejyhant size is 
preferable, as it possesses a peculiar consistency and grain. A larger 
paper is seldom required, and when the drawing to be made happens to be 
small, 2i portion of a double elephant sheet should be used. 

The paper for a colored drawing ought always to be strained upon a 
board with glue or strong gum. Before doing this, care must be taken to 
damp the face of the paper with a sponge well charged with water, in 
order to remove any impurities from its surface, and as a necessary prepa- 
ration for the better reception of the color. Tlie sponge should merely 
touch the paper lightly, and not rub it. The whole of the surface is to be 
damped, that the paper maybe subjected to a uniform degree of exj)ausion, 
thereby insuring, as it dries, a uniform degree of contraction. Submitted 
to this treatment, the sheet of paper will present, when thoroughly dry, a 
clean smooth surface, not only agreeable to work upon, but also in the 
best possible condition to take the color. 

The size of the brushes to be used will, of course, depend upon the 



33 i SHADING AND SHADOWS. 

scale to which the drawing is made. Long thin brushes, however, should 
be avoided. Those possessing corpulent bodies and fine points are to be 
preferred, as they retain a greater quantity of color, and are more manage- 
able. 

During the process of laying on a flat tint, if the surface be large — 
though this is seldom the case except in topographical drawings — the draw- 
ing may be slightly inclined, and the brush well charged with color, so that 
the edge of the tint may be kept in a moist state until the whole surface is 
covered. In tinting a small surface, the brush should never have much 
color in it, for if it have, the surface will unavoidably present coarse 
rugged edges, and a coarse uneven appearance throughout. A moderate 
quantity of color in the brush, well though expeditiously rubbed into the 
paper, is the only method of giving an even close-grained aspect to the 
surface. In fact, for mechanical drawings, there is rarely occasion for well 
charging the brush with color. The tint in the brush may be very dark 
or very light, but there should seldom be much of it. 

As an invariable rule let it be remembered, that no tint, shade, or 
shadow is to be passed over or touched until it is quite dry. 

In the examples of shading which are given in this work, it may be 
observed that all objects with curved outlines have a certain amount of 
reflected light imparted to them. It is true that all bodies, whatever may 
be their form, are affected by reflected light ; but, with a few exceptions, 
this light is only appreciable on curved surfaces. The judicious degree 
and treatment of this light is of considerable importance for the acquire- 
ment of an eifective style of shading. 

All bodies in the light reflect on those objects which surround them 
more or less light according to the situation. Wherever light extends, re- 
flection follows. If an object be isolated, it is still reached, by reflected 
light, from the ground on which it rests, or from the air which surrounds it. 

In proportion to the degree of polish or brightness in the color of a 
body, is the amount of reflected light which it spreads over adjacent ob- 
jects, and also its own susceptibility of illumination under the reflection 
from other bodies. A polished steam-cylinder, or a white porcelain vase, 
receives and imparts more reflected light than a rough casting or a stone 
pitcher. 

Shade, even the most inconsiderable, ought never to extend to the out- 
line of any smooth circular body. On a polished sphere, for instance, the 
shade should be delicately softened off just before it meets the circumfer- 
ence, and whc3n the shading is completed, the body color intended for the 
sphere may be carried on to its outline. This will give a transparency to 



335. 



PL.LXXXV^ 






d^Wk. 



k. .^^J 




'^ 







^^i^^-i: 




V 




€^ 



m^, 






^^^A^ , " -'" 



SHADING AND SHADOWS. 335 

tliat part of tlie sphere influenced by reflected liglit, wliicli it could not 
have possessed if the shade tint had been extended to its circumference. 
Very little shade should be suflered to reach the outlines even of rough 
circular bodies, lest the coloring look harsh, and present a coarse appear- 
ance quite at variance with its natural aspect. Shadows also become 
lighter as they recede from the bodies which cast them, owing to the in- 
creasing amount of reflection wdiicli falls on them from surrounding ob- 
jects. 

SJiadoics ajppear to increase in depth as their distance from the spectator 
diminishes. In nature this increase is only appreciable at considerable 
distances. Even on extensive buildings, inequalities in the depth of the 
shadows are hardly perceptible ; much less, then, can any natural grada- 
tion present itself in the shadows on a machine, which, supposing it to be 
of the largest construction, is confined to a comparatively small space. It 
is most important, however, for the eflPective representation of machinery, 
that the variation in the distance of each part of a machine from tlie spec- 
tator should at once strike the eye ; and an exaggeration in expressing the 
varying depths of the shadows is one means of effecting that object. The 
shadows on the nearest and most prominent parts of a machine should be 
made as dark as color can render them, the colorist being thus enabled to 
exhibit a marked difference in the shadows on the other parts of the ma- 
chine as they recede from the eye. The same direction is applicable in 
reference to shades. The shade on a cylinder, for instance, situated near 
the spectator, ought to be darker than on one more remote ; in fact, the 
gradation of dej)th for the shades follows that which depicts the shadows. 
As a general rule, the color on a machine, no matter what it may be in- 
tended to represent, should become lighter as the parts on which it is 
placed recede from the eye. 

Plates LXXXY. and LXXXYI. present some very good examples of 
finished shading. 

Plate LXXXY., represents, both in elevation and plan, difterent solids 
variously penetrated and intersected. The rules for the projection of these 
solids have been given under the head of Geometrical Prcjection^ and illus- 
trated in plates YI., YII., YIIL, IX., X. They are selected with a view 
of exhibiting those cases which are of most frequent occurrence, and at the 
same time elucidating the general principles of shading. 

Plate LXXXYI. presents examples of shading and shadow. 

Fig. 1 presents a hexagonal prism surmounted by a fillet. The most 
noticeable part of this figure is the shadow of the prism in the plan view. 
It presents a good example of the graduated expression which should be 



336 SHADING AND SHADOWS. 

given to all shadows cast upon plain surfaces. Its two extremities are re- 
markably different in tlieir tone. As the shadow nears the prism, it in- 
creases rapidly in depth ; on the contrary, as it approaches the other end, 
it assumes a comparatively light appearance. This difference is doubt- 
lessly a great exaggeration upon what it w^ould naturally display. Any 
modification of it, however, in the representation would destroy the best 
effect of the shadow. 

The direction which the sliades and shadows take in all the plans of 
the figures in this plate, is from the left hand lower corner. TTiis is rigor- 
ously correct, supposing the objects to remain stationary, whilst the spec- 
tator views them in both a vertical and horizontal position. J^evertheless, 
to many, this upward direction given to the shadows has an awkward 
appearance, and, perhaps, in the plan of an entire machine, the shadows 
may look better if their direction coincide with that which is given to 
them in the elevation. If, however, the shadows be correctly projected, 
their direction is an arbitrary matter, and may be left to the taste of the 
draughtsman. 

Figs. 2, 3, and 6 exemplify the complex appearance of shade and 
shadow presented on concave surfaces. It is worthy of particular notice, 
that the shadow on a concave surface is darkest towards its outline, and 
becomes lighter as it nears the edge of the object. Reflection from that 
part of the surface on which the light falls most powerfully causes this 
gradual diminution in the depth of the shadow, the greatest amount of re- 
flection being opposite the greatest amount of light. 

It may be as well to remark here, that no hrilliant or extreme light 
should be left on concave surfaces, as such lights would tend to render it 
doubtful at first sight whether the objects represented were concave or 
convex. After the body-color — which shall be treated in a subsequent 
section — ^lias been put on, a faint w^ash should be passed very lightly over 
the whole concavity. This will not only modify and subdue the light, but 
tend to soften any asperities in the tinting, which are more unsightly on a 
concave surface than on any other. 

The lightest part of a sphere (fig. 4) is confined to a mere point, around 
which the shade commences and gradually increases as it recedes. This 
point is not indicated on the figure referred to, because the shade tint on 
a sphere ought not to be spread over a greater portion of its surface than 
is shown there. The very delicate and liardly perceptible progression of 
the shade in the iirnncdiate vicinity of the light j^oint should be effected 
by means of the body-color of the sphere. If, for instance, the material 
of which tlie sphere is composed be brass, the body-color itself should be 



PL.LXXXVr. 



^36 




SHADING AND SHADOWS. 337 

lightened as it nears the light point. In like manner all polished or light- 
colored curved surfaces should be treated ; the part bordering upon the ex- 
treme light being covered with a tint of body-color somewhat fainter than 
that used for the flat surfaces. Again, if the sphere be of cast-iron, then the 
ordinary body-color should be deepened from the light point until it meets 
the shade tint, over which it is to be spread uniformly. Any curved un- 
polished surface is to be thus treated ; the body-color should be gradually 
deepened as it recedes from that part of the surface most exposed to the 
light. Considerable management is necessary in order to shade a sphere 
effectively. Tlie best way is to put on two or three softened-off tints in 
the form of crescents converging towards the light point, the first one 
being carried over the point of deepest shade. 

A ring (ng. 5) is a difficult object to shade. To change with accurate 
and effective gradation the shade from the inside to the outside of the ring, 
to leave with regularity a line of light upon its surface, and to project its 
shadow^ with precision, require a degree of attention and care in their exe- 
cution greater, perhaps, than the shade and shadow of any other simple 
figure. The learner, therefore, should practise the shading of this figure, 
as he will seldom meet with one presenting greater difficulties. 

Figs. 7 and 8 show the peculiarities of the shadows cast by a conical 
form on a sphere or cylinder. The following fact should be well noted in 
the memory : — Tliat the depth of a shadow on any object is in proportion 
to the degree of light which it encounters on the surface of that object. 
In these figures very apt illustrations of this fact may be remarked. It 
will be seen by referring to the plan (fig. 7), that the shadow of the apex 
of the cone happens to fall upon the lightest point of the sphere, and is, 
therefore, the darkest part of the shadow. So also the deepest portion ot 
the shadow of the cone on the cylinder in the plan (fig. 8) is exactly where 
it coincides with the line of extreme light. Flat surfaces are similarly 
affected, the shadows thrown on them being less darkly expressed, accord- 
ing as their inclination to the plane of projection increases. The body- 
color on a fiat surface should, on the contrary, increase in depth as the sur- 
face becomes more inclined to this plane. 

Another notable fact is exemplified by these figures : — that reflected 
light is incident to shadows as well as to shades. This is very observable 
where the shadow of the cone falls upon the cylinder. It may likewise 
be remarked, though to a less extent, on other parts of these figures. The 
reflected light on the cone from the sphere or cylinder is also worthy of 
observation. This light adds greatly to the effect of the shadows, and 
22 



338 SHADma and shadows. 

indeed to the appearance of the objects themselves. Altogether, these 
figures offer admirable scope for study and practice. 

The concentration within a small space of nearly all the peculiarities 
and effects of light, shade, and shadow, may be seen on plate LXXXYIL, 
in the examples of screws there given. 

The parts of a highly-finished colored drawing of a machine are always 
affected by a certain degree of indefinableness in their outline. 

Notwithstanding the most careful exertions of the colorist to keep 
every feature of a machine clear and distinct, some am.ount of uncertainty, 
resulting unavoidably from the proximity and natural blending of the dif- 
ferent parts, will pervade the lines which separate its component members. 
For practical working purposes, therefore, a completely colored drawing 
of a machine is unsuitable. On the other hand, a mere outline, although, 
perhaps, intelligible enough to those who are familiarly acquainted with 
the machine delineated, has an undecided appearance. As complete 
coloring renders it difficult for the eye to separate the various j)arts of a 
machine, owing to an apparently too intimate relationship between them ; 
a line drawing, on the contrary, perplexes the eye to discover any relation 
between them at all, or to settle promptly their configuration. The eye* 
involuntarily asks the question, is that part round or square, or is it even 
a distinct part of the machine at all ? As a means of avoiding the inde- 
finiteness presented by the outline in the former case, and the want of 
adequate coherence and doubtfulness in the form of the different parts 
amenable to tlie latter, recourse is not unfrequently had to a kind of semi- 
coloring, or rather mere shading of the parts of a machine. Exemplifi- 
cations of this practical style of representing machines may be observed 
in plates XL. to XLIII. inclusive, "Drawing of Machinery," pages 
204: and 206. Every figure looks complete without elaboration, and is 
clearly delineated without degenerating into the bareness of a mere skele- 
ton. Outlines and forms are at once apprehended, and every member of 
the machine is adjusted without hesitation to its proper place. 

In such drawings shading only is allowed, and therefore but slight 
scope is permitted for imparting effects ; and it is advisable to follow a 
direction previously given, and to modify the color on every part accord- 
ing to its distance from tlie eye. It may be as well also, for the purpose 
of maintaining harmony in the coloring, and of equalizing its appearance, 
to color less darkly large shades than small ones, although they may be 
situated at an equal distance from the eye. No very dark shading is per- 
missible on this species of drawing; indeed the tint should be very con- 
siderably lighter (lian on finished colored drawings. Besides presenting 



PL.LXXWIl. 



:i38 



1 




SHADING AND SHADOWS. 339 

too violent a contrast between the parts colored and those without any 
color at all, dark shading would produce, in some measure, the indistinct- 
ness which is objectionable in completely tinted drawings. 

At page 383, Plate XCYIII. is a photograph from a finished drawing 
of the engine and a boiler of the steamer Pacific. Every shadow is care- 
fully projected, every detail elaborated, and the execution perfect ; it may 
serve as a model of its class, not only for its accuracy and distinctness of 
detail, but also for its vigor as a picture. It is seldom that so much labor 
is devoted to a mechanical drawing, but the result is very satisfactory to 
the desio:ner. 

o 

FINISHED COLORING. 

Tlie coloring of drawings representing machinery requires a special 
study, the process of its development being, in many essentials, very differ- 
ent from that pursued in the artistic expression of other objects. 

The parts of a machine being usually constructed with mathematical 
accuracy, and always presenting a well-defined rigid outline, the same 
unmistakable definiteness should be maintained in any attempt to picture 
such an object on paper. There should be no "blending" of different 
colors, no doubtful finish to a tint, no softening off into the imaginative ; 
every part should present at once to the eye its form and position ; should, 
in fact, supply the place of a model of the machine. So important a fea- 
ture is this in mechanical coloring, that when correct shadows would mate- 
rially obscure any part of a machine, they should either be entirely sup- 
pressed, or, when such an omission would be very striking, so modified as 
to lessen as much as possible the obscurity thus produced. 

The color of cast-iron fresh from the foundry is commonly a very dark 
bluish black, having blended with it an almost imperceptible brownish- 
green tint or cast. To represent the casting on paper to the best advan- 
tage, the following colors should be employed : — Indian ink and indigo, 
with a very slight admixture of lake. This last ingredient is necessary ; 
for Indian ink, being actually only a very dark brown, it would, in con- 
junction merely with a blue, impart too green a cast to the tint sought to 
be realized. 

Great care should be taken in mixing these colors. First, the lake — 
crimson is preferable — should be rubbed on the pallet ; about half-a-dozen 
turns of the hand are sufficient, as too much of this color would impart a 
rusty appearance to the desired tint. The indigo may then be added, 
and lastly the Indian ink. The quantity of lake being very inconsiderable, 
about two-thirds of the mixture should be composed of Indian ink, and 



340 SHADING AND SHADOWS. 

the remaining third of indigo. This proportion, however, will be best 
ascertained by occasionally trying the tint on a scrap of drawing-paper 
during the process of mixing. When the tint appears to have approxi- 
mated as near as possible, according to the colorist's judgment, to the tint 
described above, its ingredients should be well mixed together with the 
brush. The more intimate this intermixture of the colors can be rendered, 
the better ; for if any considerable number of particles of the same color 
remain together, the tint, when essayed, will present a streaky, semi-party- 
colored appearance. The tint being thus prepared, should be left for a short 
time untouched, so as to allow the grosser particles of color to settle at the 
bottom of the saucer. 'No fear need be entertained of getting the tint too 
dark, or of mixing too much ; on the contrary, it is better to compound a 
considerable quantity and very dark in one saucer, and then gently pour 
a little into one or two others, in which, with varying quantities of water, 
different gradations of tint may be produced. The tint left in the first 
pallet should be preserved for shading or for shadows, and when it has be- 
come dry, should by no means be discarded, as it will always be service- 
able, and indeed preferable for imparting the lesser dark effects to various 
parts of the drawing. 

With one or two exceptions, which will be pointed out later, this tint, 
variously modified, is the only one to be employed for the representation 
of cast iron. It is adapted as well for expressing the shades and shadows 
as for depicting the body-color. If the shades and shadows be indicated 
by Indian ink alone, the small amount of "brilliancy" which cast iron 
naturally enjoys will disappear wherever covered with Indian ink, and 
even the effect of the body-color will be very sensibly diminished. 

The first parts of the drawing of a machine which it is usually most 
judicious to color are those of a circular form — cylinders, the more im- 
portant shafts, &c. The rods and smaller shafts, especially where they 
cross other parts of the machine, may be left until the other work is 
finished. 

Taking for granted that the learner has practised the art of shading 
according to the simple methods previously described, and, therefore, that 
he is somewhat acquainted Avith the use of the brush, let him now proceed 
to color a circular casting, it being with cast iron only that we have to do 
at present. 

Imagine this casting to be a large cylinder. First draw two faint 
pencil lines, to indicate the extremes of light and shade on its surface. 
Pass the brush, moderately full of the darkest tint, down the line of deep- 
est shade, spreading the color more or less on either side, according to the 



SHADING AND SHADOWS. 341 

diameter of tlie cylinder ; then, if possible, before this layer of tint is dry, 
towards the line of extreme liglit, beginning at the top, and encroaching 
slightly over the edge of the first tint, lay on another not quite so dark, 
but about double its width. It may be observed, that it is not very essen- 
tial to put on the second tint before the first is dry, for the latter should 
be so dark and thick, that its edges may be easily softened at any time. 
Whilst this second tint is still wet, with a much lighter color in the brush, 
proceed in the same manner with a third tint, and so on, until the line of 
extreme light is nearly attained. Repeat this process on the other side of 
the first tint, approaching the outline of the cylinder w^ith a very faint 
wash, so as to represent the reflected light wdiich ^progressively modifies 
the shade as it nears that line. Then let a darkish narrow strip of tint 
meet, and pass along the outline of the cylinder on the other side of the 
extreme line of light, after w^hich gradually fainter tints should follow, 
treated in a manner similar to that which has been already described, and 
becoming almost imperceptible just before arriving at the line of light. 

This is a very expeditious way of shading a cylinder ; but even to the 
most experienced colorist, it is not possible, by the above-described means 
alone, to impart a sufiicient degree of well-regulated rotundity to the ap- 
pearance of such an object. Superfluities and deficiencies of color will 
appear here and there. It will be necessary, therefore, to equalize to some 
extent, by a species of gross stippling, the disparities which present them- 
selves. Tliis is done by spreading a little color over the parts where it is 
deficient, and then j)assing very lightly over nearly the whole width of the 
shade, with the brush supplied with a very light wash. This process may 
be repeated to suit the degree of finish which it is desired to give the 
drawing. In the same manner the shading of all curved surfaces is to be 
treated. 

Recourse is often had to what is called " washing " or " sponging," in 
order to impart softness and circularity to certain forms. Beyond a very 
limited extent, this is a most injudicious system. It robs the shade of all 
the lightest and most brilliant particles of color, the natural position of 
which is on the surface ; it destroys that " crisp " freshness, so essential 
towards the beautiful appearance of all coloring ; and, what is still worse, 
spreads a dirty appearance not only over the whole surface of the coloring, 
but more or less on all the paper which surrounds it. Sponging should 
never be adopted, and if a slight washing with the brush be sometimes 
attempted, it should be done very lightly, and, except on rare occasions, 
not allowed to pass beyond those parts of the drawing covered with color, 
otherwise that sharp cleanly appearance, which so enhances the efibct of a 



31:2 SHADINO AND SHADOWS. 

colored drawing, will be lost. Let it, then, be remembered, that the less 
the color, whether as a shade, shadow, or tint of any kind, is touched after 
it has reached the paper, the better. The system of shading by numerous 
tints laid one over the other — a system which almost universally prevails 
— is no doubt a very easy, and, therefore, advantageous one for the initia- 
tion of beginners into a dexterous use of the brush and the grosser mysteries 
of coloring ; but no highly effective mechanical drawing can be produced 
in this manner. 

The principal shadows are the next parts of the coloring which will now 
claim attention. The outline of any shadow being drawn in pencil, along 
its inner line — the line which forms a portion of the figure of the object 
whose shadow is to be represented — ^lay on a strip of the darkest tint, wide 
or narrow, according to the width of the shadow, and then, before it is 
dry, soften off its outer edge. Tliis may be repeated as often as the taste 
of the colorist may dictate, but the color should not spread itself over 
much more than half the space occupied by the shadow. These prelimi- 
nary touches will add to the intensity of the proposed shadow, and neutralize 
a certain harshness of appearance inevitable to all shadows made equally 
dark throughout. The effect they give to the drawing is very pleasing, 
and is, moreover, quite natural, for, as previously explained, the greatest 
depth of a shadow is invariably that part of it immediately contiguous to 
the object shadowed forth. 

The representation of the casting is now to be completed by laying on 
the body-color. This might be done by a single wash of tint if the ap- 
pearance of cast iron were as light as it is usually depicted ; but its natural 
color being, on the contrary, very opaque and heavy, two and sometimes 
three washes are necessary, the first tint being rather darker than those 
which follow. Each tint should pass over the shades and shadows when 
they occur, care being taken to manoeuvre the brush at such parts very 
lightly. 

The most conspicuous fault observable in the generality of colored 

mechanical drawings is a deficiency in the depth of the tints employed. 

There appears to exist an undefinable fear of transferring to paper the 

naturally dark appearance of iron ; the result is the production of tame, 

hieffectivc representations, wliicli, instead of looking as they sliould, like 

models of iron machines, present mere faint shadows of such objects, or, 

at best, machinery constructed of some unknown, light, and rather dirty 

materials. 

The sectional surfaces of cast-iron are to bo ilidicated by one light tmt 

of indigo. 



SHADING ^VND SHADOWS. 343 

The next most extensive and important component used in the manu- 
facture of machines is wrought iron. Precisely tlie same colors are to be 
employed to represent this material as have been pointed out for cast-iron. 
The difference in the appearance of these metals is produced by altering 
the proportion of the two principal colors, — Indian ink and indigo. Tliese 
ingredients should be mixed, carefully and v^ell mixed, in about equal j)ro- 
portions, a very small quantity of crimson lake being first rubbed in the 
saucer. 

The same methods of shading and of laying on the shadows prescribed 
for cast-iron are to be adopted in the case of wrought-iron, keeping, how- 
ever, all parts of the latter lighter, particularly the body-color. The direct 
and reflected lights must also present themselves more distinctly, and to a 
much greater extent. Polished and semi-polished surfaces invariably 
afford greater contrasts of light and shade than other surfaces. The steps 
or rather glidings from one extreme to the other are, moreover, softer and 
more delicately graduated, and, therefore, greater care is requisite in repre- 
senting them on paper. These remarks are very effectively illustrated by 
the fragments of large screws shown on pi. LXXXYII. and also by the 
photograph of the steamer Pacific, Plate XCYIII. 

For the parts of wrought iron in section a light tint of Prussian blue is 
most suitable. This is the only service for which Prussian blue can pro- 
perly be made available in coloring drawings of machinery. In conjunc- 
tion with Indian ink or indigo its inherent brightness entirely disappears ; 
an ill-assorted imion with the former producing a dirty color, in appear- 
ance not unlike that presented by the surface of a stagnant pool ; and 
with the latter creating a tint bearing a striking resemblance to soiled 
glass. For mechanical drawings, then, this color must never be used in 
combination. 

Brass, except in small quantities, seldom makes its appearance in 
machinery. This is fortunate for the colorist, as there is no metal more 
difficult to represent than brass. The body-tint is composed either of 
gamboge and burnt sienna, or gamboge and crimson-lake ; the shading 
and shadows being best expressed by burnt umber. 

The most delicate and careful treatment is needed in makino^ use of 
these colors ; for, when on the paper, they are all of them very soft, and 
therefore highly sensitive to every touch of the brush. For this reason 
the shadows are best j^ut on after the body-color, otherwise their edges 
will inevitably present a smeary, indefinite appearance. 

For representing brass and copper, the method of coloring we have 
described in this section is particularly suitable. To attempt the produc- 



34:4 SHADINa AND SHADOWS. 

tion of a shade with burnt umber, by means of a succession of tints, would 
merely realize a complicated smear. We find, therefore, that the shades 
and shadows of brass are usually represented by Indian ink ; but as gam- 
boge almost invariably enters as an ingredient into the body-color of brass, 
the result is that the bright gamboge over the brown-black Indian ink ex- 
hibits a species of green, to which we cannot find any thing comparable, 
but which commonly has a very unpleasant effect to the eye. 

In shading circular surfaces great management is requisite. " "Wash- 
ing " is here entirely out of the question, for even the necessary softening 
off with the brush is attended with much difficulty. The brush should not 
pass heavily or often over the shade tint, lest unseemly deficiencies and 
streaks of color present themselves here and there, w^hich prove rather 
difficult blemishes to repair. The utmost care and experience, neverthe- 
less, cannot wholly insure the colorist against the perplexities of such 
partial failures. The only way to manage these defects is by delicate 
stippling ; suiting the depth of tint to the various degrees of shade 
affected, and then passing a soft brush, filled moderately with dark body- 
color, very lightly over the whole shade. 

A light tint of gamboge is to be used for the sections of brass. 

The directions which we have given for the most advantageous treat- 
ment of the colors representing brass, are equally applicable to those which 
exhibit the nearest approach to copper, the colors to be used for this metal 
opposing nearly an equal amount of difficulty in their management. A 
mixture of orange chrome and lake, or red-lead and lake, best represent 
this metal ; its shades and shadows being indicated by sepia, whilst its 
sectioning is shown by a light tint of orange chrome. 

Such are the colors, and such is the manner of treating them, em- 
ployed for depicting on paper each of the principal metals used in ma- 
chinery. 

Having explained in detail the tinting of machinery in reference both 
to its sliading and body-color, we propose to complete our remarks on 
mechanical coloring with a few suggestions for imparting some peculiar 
effects to the rej)resentations of masses of machinery. 

"We have already noticed that tlie shades and shadows of a machine 
are modified in intensity as their distance from the eye increases. Its 
body-color should be treated in a similar maimer, becoming lighter and 
less bright as the parts of the machine which it covers recede from the 
spectator. 

When the large circular members of a machine have been shaded, the 



SHADING AND SHADOWS. 345 

shadows, and even tlio body-color on those parts furthest removed from 
the eye, are to follow, and the proportion of Indian ink in the tint nsed 
shonld increase as the part to be colored becomes more remote. A little 
washing, moreover, of the most distant parts is allowable, as it gives a 
pleasing appearance of atmos];)heric remoteness, or depth, to the color thns 
treated. 

TKe amonnt of light and reflection on the members of a machine shonld 
diminish in intensity as the distance of snch objects from the spectator in- 
creases. As it is necessary, for effect, to render, on those parts of a ma- 
chine nearest the eye, the contrast of light and shade as intense as possible, 
so, for the same object, the light and shade on the remotest parts shonld 
be snbdued and blended according to the extent or size of the machine. 

A means of adding considerably to the definiteness of a colored me- 
chanical drawing, and of promoting, in a remarkable degree, its effective 
appearance, is obtained by leaving a very narrow margin of light on the 
edges of all surfaces, no matter what may be the angles which they may 
form with the snrfaces that join them. Tliis shonld be done invariably ; 
we do not even except those edges which happen to have shadows falling 
on them ; in snch cases, however, this margin, instead of being left qnite 
white, which would have a harsh appearance, may be slightly snbdued. 
Tlie difficulty of achieving this effect, of imparting a clear, regular, un- 
broken appearance to these lines of light, seems very formidable, and, 
indeed, nearly insuperable. The hand of the colorist may be as steady 
and confident as a hand can be, and yet fail to guide the brush, at an 
almost inappreciable distance from a straight or circular line, with that 
precision and sharpness so requisite for the accurate delineation of this 
beautiful effect. "We shall, however, explain a novel and effective method 
of arriving at this most desirable result. 

Suppose the object about to receive the color to be the elevation of a 
long flat rod or lever, on the edge of which a line of light is to be left. 
Fill the drawing pen as full as it will conveniently hold with tint destined 
to cover the rod or lever, and draw a broad line just within, but not touch- 
ing, the edge of the lever exposed to the light. As it is essential for the 
successful accomplishment of the desired effect that this line of color should 
not dry, even partially, until the tint on the whole side of the lever has 
been put on, it will be as well to draw the j)en again very lightly over 
the same part, so that the line may retain as much tint as possible. Im- 
mediately this has been done, the brush, properly filled with the same 
tint, is to pass along and join the inner edge of this narrow strip of color, 
and the whole surface of the lever filled in. Thus a distinct and regular 



346 SHADING AND SHADOWS. 

line of light is obtained, and, in fact, the lever, or whatever else the ob- 
ject may be, covered in a shorter time than usual. A still more expedi- 
tious way of coloring such surfaces is to draw a second line of color along 
and joining the opposite edge of the lever or other object, and then expe- 
ditiously to fill in the intermediate space between the two wet lines, by 
means of the brush. In this manner a clear imiform outline to the tint is 
obtained, which could not be effected in any other way. As celerity in 
the movements of the colorist is very necessary to carry out properly this 
method of leaving a light edge to the boundaries of flat surfaces, and as 
confidence in possessing the requisite ability to perform it must precede 
success, a little practice is desirable before essaying it on any drawing 
of importance. The blades of the drawing pen must not be sharp, and the 
pien should be used with great precaution and delicate lightness, otherwise 
the blades will cut more or less the paper and leave their course visible — 
an unsightly betrayal of the mechanical means employed to obtain such 
regularity in the coloring. Flat circular surfaces may be treated in the 
same manner, by using the pen-compass in place of the drawing pen. 
When such surfaces are rather extensive, it will be judicious to color them 
in halves, or in quadrantal spaces, taking great care,, when joining the 
parts together, that they may overlap or fall short of each other as little 
as possible. The appearance of these junctions may be obliterated by 
slightly washing them, or by going over the whole surface wdth a very 
light tint, and, in passing, gently rubbing the seams wdth the brush. By 
similar means the line of light on a cylinder, shaft, or other circular body, 
may be beau.tifully expressed. To indicate this light with perfect regu- 
larity is highly important, for if a strict uniformity be not maintained 
throughout its whole length, the object will look crooked or distorted. 
After having marked in pencil, or guessed the position of the extreme 
light, take the drawing pen, w^ell filled with a just perceptible tint, and 
draw a line of color on one side the line of light, and almost touching it ; 
then with the brush, filled with similar light tint, join this line of color 
whilst still wet, and fill up the space unoccupied by the shade tint, within 
which the very light color in the brush will disappear. Let that part of 
the object on the otlier side of the line of light be treated in the same way, 
and the desired efiect of a stream of light clear and mathematically regu- 
lar will be obtained. The effectiveness and expedition of this method will 
be most obvious in coloring long circular rods of small diameter, where 
the want of accuracy is more immediately perceptible. The extreme 
depth of shade, as well as the line of light in such rods may, with great 
effect, be indicated by filling the pen with dark shade tint, and drawing it 



SHADING AND SHADOWS. 347 

exactly over the line representing the deepest part of the shade. On 
either side and joining this strip of dark color, anotlier, composed of 
lighter tint, is to be drawn. Others successively lighter are to follow, 
until, on one side, tlie line of the rod is joined, and on the other the 
lightest part of the rod is nearly reached. The line of light is then to be 
shown, and the faint tint used on this occasion spread wdth the brush 
lightly over the whole of that part of the rod situated on either side of 
this line, thus blending into smooth rotundity the graduated strips of tint 
drawn by the pen. 

In all tinted drawings the more important parts, whether the machinery 
or the structure, should be more conspicuously expressed than those parts 
which are mere adjuncts. Thus, if the drawing be to explain the construc- 
tion of the machine, the tint of edifice and foundations may be kept lighter 
and more subdued than those of the machine ; and if the machine, on the 
contrary, be unimportant, it may be represented quite light, or in mere 
outline, whilst the edifice is brought out conspicuously. 

As has been stated, there are two methods of shading, by flat tints and 
by softened tints; but in the w^ork, the "Engineer and Machinist's Draw- 
ing Book," from which the preceding Treatise on Shading and Shadows 
has been taken, the process of coloring by flattened tints, or superposition 
of tints, is ignored, and the method confined to that of softened tints ; and 
very strong objection is made to washing, although '' a little washing of the 
most distant parts is allowable" (page 345). By this process recommended 
for coloring, a distinct and even an artistic drawing of architectural or 
mechanical objects could undoubtedly be made by a skilful draughtsman ; 
but by the inexpert, we think that the process of coloring by flat tints will 
be found much the more simple and readier way of producing a respect- 
able drawing ; and the method given pages 330 and 331 applies equally 
well to drawings in color. 

With regard to washings, the soft sponge is an implement not to be 
neglected by the draughtsman; it is an excellent means of correcting 
great errors in drawing, better than rubber or an eraser, but care of course 
nmst be taken to wash and not to rub off the surface, and for errors in 
coloring washing is almost the only corrector. In removing or softening 
color on large surfaces, the sponge is to be used, and for small spots the 
brush. Whilst coloring, keep a clean, moist brush by you : it will be ex- 
tremely useful in removing or modifying a color. 

The immediate effect of washing is to soften a drawing, an effect often 
very desirable in architectural and mechanical drawings, and the process is 
simple and easily acquired ; keep the sponge or brush and water used 



348 SHADING AND SHADOWS. 

clean ; after the washing is complete take up the excess of moisture by the 
sponge or brush, or by a piece of clean blotting paper. Where great 
vigor is required, let the borders of the different tints be distinct ; if the 
strips are narrow, the effect in comparison with that obtained by softened 
tints is as a line engraving compared with a mezzotint. 

With regard to the colors to be used to represent different materials, 
there are no conventional tints, none that draughtsmen have agreed upon 
to be uniformly used, and we think that some improvement can be made 
on those before recommended. India ink, it has been observed, is not a 
black, but a brown, making with a blue a greenish cast, and with gamboge 
a smear. A colored drawing is better without the use of India ink at all ; 
any depth of color may be as well obtained with blue as with black ; there 
is also an objection to gamboge, that it is gummy, and does not w^ash well, 
and the effect is better obtained with yellow ochre. For the reds, the mad- 
der colors are the best, as they stand washing. 

For" the shade tint of almost every substance a neutral tint, Payne's 
grey, or madder brown subdued with indigo ; for the local color, or what 
has improperly been designated as the body color, for cast-iron use a wash 
of indigo, and for wrought-iron, of cobalt blue ; for sections of these sub- 
stances, pure indigo for cast-iron, and cobalt blue for wrought-iron, with 
hatchings of deep tints of the same color. For shadows on brass or cop- 
per use madder brown ; local color for brass, yellow ochre or Indian yel- 
low ; and for copper, a light wash of Yenetian or light red ; for sections, 
pure tints or washes, with deep hatchings of the same. 

For building material, as granite or brick, imitate the color, but float 
on the tints, leaving it in patches, and soften by washes of clean water, or 
by some local tints which suit the material. The outer walls of houses, in 
section, are often colored in a simple tint of carmine or madder brown. 
For wood of a light color use a tint of burnt sienna ; for dark woods, a 
mixture of burnt umber and sepia ; and for the shadows madder brown. 

Plates LXXXYIII. and LXXXIX. are illustrations in chromo-lithogra- 
phy from colored drawings. It has not been possible to express the effect 
given by hand, but they may serve in a measure as models, with the text 
as a guide. Every one wishing to become a draughtsman should make a 
scrap-book or collection of such drawings as he can from time to time pick 
up, to serve him as guides, study the effects which are given in water-color 
drawings ; in the architectural department especially we know of nothing 
cheaper and better than the illustrations of some of the London papers ; 
whether in ink or color, they afford capital studies of design and of execu- 
tion. 



PLATE LXXXYli. 




Am, fholo- Lithographic, Cfl.N.Y.(0sborn»»Frcc«S5] 



TE L XXXIX 






S49 



PLATE XC. 



J 


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111 I'l lir III 'II II' I'l IK III III Ml <l> III III Ml III ,i| 11 III III M; .|| Mi III "1 


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Ml IP I'l III <ll II' III I'l 1" III "1 III I'l M' II' VI 'II Ml Ml <l< <l' <!/ M' <li 


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a III III MI III III I'l m \^, VI VI (1 i;i oi m -ii mi u hi m M' i'i m •!' 'i' 




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V V. Ill 01 i» W 111 <i 1" 'II ,'l « * ij' « 01 'll '1' I'l .'' -1' 'I '1' '" '" 





i: # t ^ A ^ ^ 


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1. 1. ^ t t. ft 4 




TOPOGRAPHICAL DRAWING. 349 



TOPOGKAPHICAL DEAWING. 

Topographical Drawing is the delineation of the surface of a locality, 
with the natural and artificial objects, as houses, roads, rivers, hills, etc., upon 
it in their relative dimensions and positions ; giving, as it were, a miniature 
copy of the farm, field, district, etc., as it would be seen by the eye moving 
over it. Many of the objects thus to be represented can be defined by 
regular and mathematical lines, but many other objects, from their irregu- 
larity of outline, it would be very difficult thus to distinguish ; nor are 
the particular irregularities necessary for the expression. Certain con- 
ventional signs have therefore been adopted in general use among drafts- 
men, some of which resemble, in some degree, the objects for which they 
stand, whilst others are purely conventional. Tliese signs may be ex- 
pressed by lines, or by tints, or by both. We commence with those in 
lines, and in the latter j)art of our treatise, finish with examples in color. 

Plate XC. fis:. I, represents meadow or grass line, the T" -""^ .^w"'- — ^t- 
short lines being supposed to represent tufts of grass ; -''%!^-''^'^'i:,^-'''%l^^t'^ 
the base line of these tufts should always be parallel to^>*""-'''%..trX-"'- 
the base of the drawinor, no matter what may be the ;,\w."'''!:I^''C%i<^^'-*'^ -*■ 
shape of the enclosure. Fig. 1 expresses the same thing ^//-^ ■^'"- -^.t'^-"^'-- ->^"- 
on a larger and coarser scale. 

Fig. II. represents an orchard ; fig. III. a forest or clump of forest trees. 
In both these examples, the trees are represented in elevation ; this is a 
very common method of representation, but not consonant w^ith the other 
parts of the plan. It seems better that trees should be represented in 
plan, as in fig. TV. Orchards may be represented thus (fig. 2), and forests, 
on a larger scale, by a sort of distinctive foliage, according to the kinds of 
trees ; thus fig. 3 may represent chestnut, fig. 4 oak, fig. 5 pine and fir. 
When trees occur upon a hill-side, the shading lines of the hill-side should 
be interrupted to receive the body of the tree, but not its shadow, which 
may be drawn independently of them when the slope is slight, but when 



350 



TOPOGRAPHICAL DRAWING. 



it is steep the shadows may be omitted, and the trees shaded nearly as 
dark as that of the slope, but the foliage should be represented rather 
sparse. 











Fig. 4. 





Fig. 5. 



Fig. Y. represents a house and cultivated ground ; the walks and roads 
are in white, the buildings are marked by diagonal lines. The cultivated 
land by parallel rows of broken and dotted lines, supposed to be furrows. 
Sometimes signs are used to represent the crops. 

Fig. YI. represents marsh land, v/ater and bog. Fig. YII., a river 
with mud and sand banks. Sand is represented by fine dots made with 
the point of the pen ; mud in a very similar way, but the dots should be 
much closer together. Gravel is represented by still coarser dots, and 
stones by irregular angular forms, imitating their appearance, as seen from 
above. 

Fig. YIII. represents a bold shore boimded by cliffs. Water is almost 
invariably represented in the same way, except in connection with bogs 
(fig. YI.), by drawing a line parallel to the shore or coast, following its 
windings and indentations, and as close to it as possible; then another 
parallel a little more distant, a third still more so, and so on. Small ponds 
are sometimes represented by parallel horizontal lines, but usually by the 
curved lines of shore. Brooks, and even rivers, when the scale is small, 
are represented by one or two lines. The direction of the current is shown 
by arrows. 

Fig. 6 represents a turnpike. If the toll-bar and marks for a gate 
be omitted, it is a common highway. Fig. 7 represents a road as sunk or 
cut through a hill. Fig. 8, one raised upon an embankment. Fig. 9 is a 



3SW, Bar 

£m 



lyih^N . 



FiL'. G. 



Fig. 7. Fig. 8, 



Mimiiiiimiii i i iii r 

Fig. 9. 



railroad, often represented without the cross-tie, by two heavy parallel 
lines, sometimes by but one. 



TOPOGRAPHICAL DRAAVING. 



351 



Fig. 10 represents a bridge with a single pier. Fig. 11, a swing or 



JLJ ILL -^ 



r 




Fig. 10. 



Fig. 11. 



Fig. 12. 



Fig. 13. 



Fig. 14 



mm 




Fi- 15. 



Fig. 16. 



draw bridge. Fig. 12, a suspension bridge, and 
iig. 13 a ford. Fig. 11:, a lock of a canal. Ca- 
nals are represented like roads, except that in 
the latter the side from the light is the shaded 
line, in the former, the side to the light. 

Fig. 15 represents dwellings, or edifices of 
any sort ; they are often made distinctive of their purpose by some small 
prefix, as a pair of scales for a court-house, an elevation of a sign-post for 
a tavern, a letter for a post-office, a horseshoe for a smithy, a small water- 
wheel for a water-mill, and a chimney for a steam-mill. 

Fig. 16 represents a church or cathedral ; this is sufficiently expressed 
by its plan ; but usually, churches are rej^resented according to their own 
plan, with the distinctive prefix of a cross or a steeple. 

Tlie localities of mines may be represented by the signs of the planets, 
which were anciently associated with the various metals, and a black circle 
for coal. Thus ^ Mercury, $ Copper, "^ Lead, ]) Silver, O Gold, $ Iron, 
U Tin, ® Coal. 

Ofi the Representation of HUU. — The two methods in general use for 
representing with a pen or pencil the slopes of ground, are known as the 



^~- 



m^Milii 



4i 





Fi-. 13. 



OK 9. 



TOPOGEAPHICAL DRAWING. 



vertical and the horizontal. In the first (fig. IT), the strokes of the pen 
follow the course that water would take in running down these slopes. 
In the second (fig. 1 8), they represent horizontal lines traced round them, 
such as would be shown on the ground by water rising progressively by 
stages, 1, 2, 3, 4, 5, 6, up the hill. The last is the most correct represen- 
tation of the general character and features of the ground, and when ver- 
tical levels or contours have been traced by level at equal vertical dis- 
tances over the surface of the ground, they should be so represented ; or 
when, by any lines of levels, these contours can be traced on the plans 
with accuracy, the horizontal system should be adopted ; but where, as in 
most plans, the hills are but sketched in by the eye, the vertical system 
should be adopted, it affords but proximate data to judge of the slope, 
w^hereas, by the contour system, the slope may be measured exactly. It is 
a good maxim in topographical drawing, not to represent as accurate any 
thing which has not been rigorously established by surveys. On this 
account, for general plans, when the surface of the ground has not been 
levelled, nor is required to be determined with mathematical precision, 
we prefer the vertical to the horizontal system of representing slopes. 

On drawing hills on the vertical system, it is very common to draw 
contour lines in pencil as guides for the vertical strokes. If the horizontal 
lines be traced at fixed vertical intervals, and vertical strokes be drawn 
between them in the line of quickest descent, they supply a sufficiently ac- 
curate representation of the face of the country for ordinary purposes. It 
is usual to make the vertical strokes heavier the steeper the inclination, 
and systems have been proposed and used, by which the inclination is 
defined by the comparative thickness of the line and the intervening 
spaces. 

In describing ground with the pen, the hght is generally supposed to 
descend in vertical rays, and the illumination received by each slope is di- 
minished in proportion to its divergence from the plane of the horizon. 

Thus in fig. 19, it will be seen 
that a horizontal surlace receives 
an equal portion of light with the 
inclined surface resting upon it, 
and as the inclined surface is of 
ri„ 19, greater extent, it will be darker 

than the horizontal in proportion to the inclination and consequent in- 
crease of the surface, and on this principle varied forms of ground are 
represented by proportioning the thickness of stroke to the steepness of 
tlie slope. 




TOPOGRAPHICAL DRAWING. 



353 



In tlie German system as proposed by Major Lelimann, of representing 
the slopes of ground by a scale of shade, the slope at an angle of 45°, as 
reflecting its light horizontally, is supposed to be the greatest ever required 
to be shown, and is represented by black, whilst the horizontal plane re- 
flecting all rays upward is represented by white, and the intermediate 
slopes by difl'erent j^roportions of black in the lines to white in the spaces 
intervening. We have not thought it necessary to give an illustration of 
this scale of shade, as it does not discriminate between slopes of greater 
inclination than 45°, preferring the modification as proposed for the U. S. 
Coast Survey, adapted to the representation of all necessary slopes, and 
consonant ^\ ith the demonstration, fig 19. Fig. 20 represents this scale of 




Fig. 20. 



Slope. 


\ 

Proportion of 
Black, j White. 


2i^ or 2^3 


1 


10 


5= or 63 


2 


9 


lOO or 110 


3 


8 


150 or 160 


4 


7 


250 or 260 


5 


6 


350 


6 


5 


450 


7 


4 


60O 


S 


3 


750 


9 


2 ' 



shade tabellated, the following are the proportions of black and white for 
difl'erent inclinations, and the construction may be 
easily understood from fig 20. Tlius form eight paral- 
lel rectangles according to the number of slopes to 
be represented ; divide each of these rectangles into 
eleven parts, then the proportion of white to black in a 
slope of 2^° will be to make one of these parts black ; 
of 5° two parts, of 10° three, and so on. Kow thicken 
the lines according to this proportion, and copy the 
strokes till the hand becomes habituated to their for- 
mation, and the eye so practised, that the graduation for all practical pur- 
poses may be performed without direct reference to the scale. 

0?i Drawing Hills l)ij Contours. — Draw first the curves which have 
been traced on the ground by levels, and these should be distinguished 
from the other lines by color, as red, or by size of lines. It should be ob- 
served that whatever point has been actually established by survey, it 
should not be confounded with sketching by eye. If there are no such 
lines, but it is merely intended to sketch the hills as in the usual vertical 
style, lay off the curves at equal vertical intervals, say 10, 20, 50 or 100 

feet, according to the scale, and then proceed to fill in. Tlie ground between 
23 



354 



TOPOGEAPHICAL DRAWING. 



these fixed curves or sections, is supposed to slope Tiniformly. Divide the 
space therefore equally, and draw within each set of curves as many lines 
as may be suited to the scale of the map, and the vertical intervals be- 
tween the curves. Draw the lines with firmness, and let them have a 
length varying from one to three fourths of an inch, according to the great- 
er or less degree of the slope. When the hill is steep the lines should be 
short and heavy, growing longer and lighter as the inclination becomes 
less. The lines should nearly touch each other, so as to appear almost 
consecutive, but not overlap, nor with a determinate interval between 
their ends. Fig. 21 represents the half of the hill, fig. 18, and at double 
scale, completed by drawing the intermediate contour lines. 




Fig. 21. 

Drawing hills by contours is of comparatively late introduction, and 
is generally practised abroad, but little used here ; it is more difiicult for 
the draughtsman, and no more expressive of the features of the ground 
than the vertical system, and has little to recommend except where actual 
lines have been traced, and it becomes a record of facts. Certain lines 
in pencil are necessary for the proper drawling according to the vertical 
system, but when the drawing is complete, an implied line is merely left. 
Hills are much more effectively expressed by the brush than the pen, and 
much more readily, of w^hicli illustrations w^ill be given further on. 

In our list of conventional signs we have given but few, and these 
only the most prominent. It is useless to tax the memory with many, as 
the purposes for which an edifice or locality is intended will supply some 
characteristic by which they are easily distinguished ; as in case of mills 
already given, or as in case of a graveyard by a tombstone, a quarry by 
a stone-hammer, a battle field by crossed swords, &c. When there is no 
obvious characteristic, the positions may be lettered or numbered and ex- 
plained by marginal notes, if tlierc be not room on the plan in its appro- 
priate locality. 



I 



TOPOGRAPinCiiL DRAWING. 355 



PLOTTING. 



Plotting is the making of the plan on paper from the measurements 
taken in the field. 

The rough sketch is usually made in the field-book, that is, the book 
kept in the field, in which all the steps or observations of the survey are 
noted on the spot. The field-book is generally ruled with a middle col- 
umn, from one half to one inch in width. This middle column is intended 
to represent the station line itself, and all lines crossing the station line 
are not drawn directly across the middle column, but arrive at one side 
and leave it on the other, at points precisely opposite. The middle colimin 
is reserved entirely for the angles and measures, made in direction of the 
station line. All measurements of offsets or angles other than those on 
the direct line are entered in the marginal spaces at each side of the middle 
column, according to the side of the station line on which they are taken. 
The stations are marked thus O, and the notes commence at the bottom of 
the page. 

Scales. — The choice of the scale for the plot depends in a great measure 
on the purpose for which the plan is intended. It should be large 
enough to express all the details wdiich it is desirable, modified by the cir- 
cumstances, whether the map is to be portable, or whether space can be 
afforded for the exhibition of a large plan. We must adapt our plan for 
the purposes which it is intended to illustrate, and the place it is to oc- 
cupy. 

Plans of house lots are usually named as being so many feet to the 
inch ; plots of farm surveys, as so many chains to the inch ; maps of sur- 
veys of States, as so m^any miles to the inch, and maps of railway surveys, 
as so many feet to the inch, or so many inches to the mile. 

For farm surveys, if of small extent, two chains to the inch is a con- 
venient scale ; for larger farms three chains to the inch. Tliis last scale is 
that prescribed by the English Tithe Commissioners for the first-class 
maps. One acre laid out in the form of a square, to the scale of 

1 chain to the inch occupies 3.16 inches square. 
1^ " « " 2.10 " " 

2 « " " 1.58 " " 

3 " " " 1.05 " " 

and so on. 



356 TOPOGKAPHICAL DE AWING. 

Knowing liow mncli is the area of tlie ground to be plotted, if tlie form 
is square we can easily determine the side of the square occupied, by mul- 
tiplying the square root of the area in acres by 3.16, and dividing the 
product by the number of chains to the inch in the scale assumed. Thus 
if 50 acres were to be plotted in a square, to the scale of 3 chains to the 

inch = V50 = 7.07. 7.07 x 3.16 = 22.34. -^ = 7.45 inches, side 

of the square of the plot on a scale of 3 chains to the inch. This rule will 
assist the draughtsman in selecting a scale for figures not very irregular in 
form. 

State surveys are of course plotted on a smaller scale than those of 
farms. On the U. S. Coast survey all the scales are expressed fractionally 
and decimally. The original surveys are generally on a scale of one to ten 
or twenty thousand, but in some instances the scale is larger or smaller. 
The public surveys embrace three general classes: — 1. Small harbor 
charts. 2. Charts of bays, sounds, &c. 3. General coast charts. 

The scales of the first class vary from 1 : 5,000 to 1 : 60,000, according 
to the nature of the harbor and the difi'erent objects to be represented. 

The scale of the second class is usually fixed at 1:80,000. Prelimi- 
nary charts are, however, issued of various scales, from 1 : 80,000 to 
1 : 200,000. 

Of the third class the scale is fixed at 1 : 400,000 for the general chart 
of the coast from Gay Head to Cape Henlo]3en, although considerations 
of the proximity and importance of points on the coast may change the 
scales of charts of other portions of our extended coast. 

On all plots of large surveys, it is very desirable that the scales adopted 
should bear a definite numerical proportion to the linear measurement of 
the ground to be mapped, and that this proportion should be expressed 
fractionally on the plan, even if the scale be drawn or expressed some 
other way, as chains to the inch. The decimal system has the most to 
recommend it, and is generally adopted in government surveys. 

For Eailroad Survevs, the New York o-eneral railroad law directs the 
scale of map which is to be filed in the State engineer's office, to be 500 
feet to one-tenth of a foot, 1 : 5,000. 

For the Canal Maps, a scale of 2 chains to the inch, 1 : 1584 is em- 
ployed. In England, plans and sections for projected lines of inland com- 
munication, or generally for public works requiring the sanction of the _ 
Legislature, are required, by the "standing orders," to be drawn to scales^' 
not less than 4 inches to the mile, 1 : 15,840, for the plan, and 100 feet to 
the inch, 1 : 1,200, for the i)rofiles. 



TOPOGRAPUICAL DRAWING. 357 

In the United States engineer service, the following scales are pre- 
scribed : 

General plans of buildings, 10 feet to the inch, 1 : 120 

Maps of ground -with horizontal cun-es 1 foot apart, 50 " " 1 : GOO 

Topographical maps 1^ miles square, , . .1 mile to 2 feet, 1 : 2,G40 

" " comprising 3 miles square, .1 " 1 foot, 1 : 5,280 

'' " " bet. 4 and 8 miles, 1 " G in., I : 10,5G0 

" " " 9 miles square, .1 " 4 » 1 : 15,840 

Maps not exceeding 24 miles square, . . .1 " 2 " 1 : 31,680 

" comprising 50 " " . . .1 *' 1 " 1 : G3,360 

" 100 " « . . .1 " i " 1 : 126,720 

Surveys of roads and canals, . . . .50 feet to 1 " 1 : 600 

The description of various scales and the principles of their constrnc- 
tion will be found at pp. 17 and 18, to which the reader is referred. 

In plotting from the field book, the first lines to be laid do^vn are the 
outlines or main lines of the survey. If the survey has been made by 
triangles, the principal triangles are first laid down in pencil by the inter- 
section of their sides, the length being taken from the scale and described 
with compasses ; if the lines are longer than the reach of the compasses, 
or the extent of the scale, lay off the length on any convenient line, and 
measure and describe with beam compasses. The j^rincipal triangles being 
laid down, other points will be determined by intersections in the same 
manner as measured on the ground. In general, when the surveys have 
been conducted without instruments to measure the angles, as the compass 
or theodolite, the position of the points on paper are determined by the 
intersection and construction of the same lines as has been done in the 
field. 

Surveys are mostly conducted by measm-ing the inclination of lines to 
a meridian or to each other by the compass, or by the theodolite. In the sur- 
veys of farms, where great accuracy is not required, the compass is most 
used. The compass gives the direction of a line in reference to the magne- 
tic meridian. The variation from the true meridian, or a direct north and 
south line, varies considerably in different parts of the country. In 1840, 
the line of variation in which the needle pointed directly north, passed 
in a nearly straight direction from a little west of Cape Hatteras, 
i^. C, through the middle of Virginia, about midway between Cleve- 
land, Ohio, and Erie, Pa., and through the middle of lakes Erie and 
Huron. At all places east of this line the variation is westerly, that is, the 
needle points west of the line north. West of this line the variation is 
easterly. 



358 



TOPOGRAPHICAL DEAWING. 



Fig. 

-ay- 

3.23 

CO 

-(5)- 
3.54 

p4 



-i^y- 

2.22 

CO 

-(3)- 
1.29 



12; 

2.78 

f4 

CO 
Fig. 22. 



22 represents the field notes of a survey by compass. Fig. 23 a 
plot of tlie same, witli the position of tlie protractor in laying 
off the angles. In this way of plotting, a meridian is laid off 
at the intersection of each set of lines. Sometimes the angles 
are plotted directly from the determination of the angle of de- 
flection of two courses meeting at any point, without laying 
down more than one meridian: Figure 24. "When the first 




Fis. 23. 



letters of the bearing are alike, that is, both I^. or both S., and the last let- 
ters also alike, both E. or both W., the angle of deflection C B B' will be 
the difference of the bearings, or, in this instance, 20°. 





VJ^ 



Fig. 25. When the first letters are alike and the last different, the 
angle C B B' will be the sum of the two bearings. 



TOPOGRAPHICAL DRAWING. 



359 



Fig. 26. When the first letters are different and the last alike, sub- 
tract the sum of the bearings from 180° for the angle C B B' : Avhen both 
the first letters and last are different, subtract their difference from 180° 
for the angle. 

Instead of drawing a meridian through each station, or laying off the 
angle of deflection, by far the easiest way is to lay off but a single meridian 
near the middle of the sheet ; lay off all the bearings of the survey from 
some one point of it as shown in fig. 27, and number to correspond 
with the stations from which 
the bearings are taken, and 
then transfer them to the 
places where they are wanted 
by any of the instruments used 
for drawing parallel lines. For 
the protracting of the rough 
plan, sheets of drawing paper 
can be bought with protractors 
printed on them. When the 
plans are large, it is often 
convenient to lay out two or 
three meridians on different 
parts of the sheet and lay off 
the bearings of lines adjacent to each meridian upon them. 

In plotting from a survey by a theodolite or transit, it is generally 
usual to lay off the angles of deflection of the difiereiit lines as taken 
in the field, plotting all the tie lines as corrections. 

When the plot of a survey does not close, that is, come together, or 
return to the point of commencement, as it seldom does exactly, it may 
be corrected or forced ; but first be sure that the bearings and distances as 
given in the field book are laid down accurately, and then proceed to cor- 
rect as follows ; thus, let A B C D E, 
fig 28, be the boundary lines plotted 
according to the notes, and suppose 
the last course comes to E instead 
of ending at A as it should. Sup- 
pose also that there is no reason to 
suspect any error more in one line 
than another, that the measures 
and bearings of all are equally cer- Fig. 28. 





Ac< 



360 



TOPOGRAPHICAL DRAWING. 



tain; then the inaccuracy must be distributed among all the lines in 
proportion to their length. Each point, B, C, D, E, must be moved in a 
direction parallel to E A, by a certain distance. Thus add together the 
length of all the lines, and this sum is to the line A B, as the error A E is 
to the correction B B' ; for the next point, the whole sum is to A B, B C, as 
the error is to the correction, E C ; and so on ; obtaining the second term 
of the proportion by adding consecutively the different lines. This calcu- 
lation may be much simplified by the use of the sector, according to the 
rule given for finding a fourth proportional (p. 23). Take the error A E 
from the plan, and open the sector until this quantity, becomes the trans- 
verse distance of the first term or sum of the lines ; then the distance be- 
tween the points corresponding to the consecutive sums will be the corre- 
sponding error. 

The best way of correcting errors and of plotting a survey, whether 
made by compass or by theodolite, is by balancing the survey, or correct- 
ing the latitudes and departure of the courses so that they shall be equal. 
Eor the method of doing this, we refer to any of the recent works on sur- 
veying. From Gillespie on Land Surveying, we have taken most of the 
preceding on the plotting of angular surveys, and the following para- 
graph on balancing. 






Total Latitude 


Total Departure 


Sta. 


from Sta. 1. 


from Sta. 1. 


1 


0.00 


0.00 


2 


-f 2.21 K 


+ 1.55E. 


3 


+ 2.36 N. 


+ 2.S3 E. 


4 


+ 1.15 K 


-f 4.69 E. 


5 


- 1.78 S. 


+ 2.69E. 


1 


0.00 


0.00 



This table represents the 
survey as given, ^g. 22, bal- 
anced. 

To plot from this table, 
draw a meridian through the 
point taken for station 1, as 
in fig. 29. Set off upward 
from this along the meridian 
the latitude 2.21 chains north 
to A, and from A to the right or E. set off the departure 1.55 chains. This 
gives the point or position of Station 2, join 1 and 2. From A again set off 
upward 2.3C chains, and from B to the right perpendicularly set off 
2.83 chains, wliich gives position of Station 3, join 2 and 3 : and so pro- 
ceed setting off North latitudes upwards, and South downwards, East 



Fig. 29. 



TOrOGRAPHICAL DRAWING. 



3G1 



departures perpendicularly to the right, and West perpendicularly to tlie 
left. 

In balancing surveys made by a theodolite, a meridian is assumed, 
generally one of the lines of tlie survey. The most convenient will be 
some long line of which the survey lies all to one side. 



^^^ 




Fis 30. 



Tlie advantages of this method of plotting are ^^^^^^ 
its accuracy, rapidity, the impossibility of an er- 
ror in one point affecting the others, and the cer- 
tainty of coming together. 

The above explains the method of plotting the 
main lines of the survey ; the filling in is from 
points established from these main lines, either 
by the construction of triangles, by measure, or 
by angles, or by perpendiculars. In case of un- 
important lines, as the crooked brook for instance, 
fig. 30, offsets are taken to the most prominent 
angles, as, a^ «, a^ and the intermediate bends are 
sketched by eye into the field book. In copy- 
ing them on the plan a similar construction is 
adopted. 

The most rapid way of plotting the offsets, is 
by the use of a plotting and offset scale, fig. 31 — 
the one being fixed parallel to the line A B from 
which the offsets are to be laid off, at such a dis- 
tance from it, that the zero line on the movable 
scale coincides with it, whilst the zero of its own 
scale is on a line perpendicular to the position of 
the station A from which the distances were meas- 
ured. It is to be observed that in the field book 
all the measures are referred to the point of be- 
ginning on any one straight line. Having placed the plotting scale, move 
the offset scale to the first distance by the scale at which an offset has 
been taken, mark off now on the offset scale the length of the offset on 




362 TOPOGRAPHICAL DRAWING. 

its corresponding side of tlie line. Proceed then to the next distance, es- 
tablishing thus repeated points, join the points by lines as they are on the, 
ground. 

The plotting and offset scale must of course be of the same scale as the 
rest of the drawing, on which account it may not always be possible to 
obtain such scales adapted to those of the plan ; but they may be easily 
constructed of thick drawing paper or pasteboard. 

When a great deal of plotting to one scale is necessary, as in govern- 
ment surveys, the offset scale may be made to slide in a groove upon the 
plotting scale. 

In protracting the triangles of an extended trigonometrical survey in 
which the sides have been calculated or measured, it is better to lay down 
the triangles from the length of their sides rather than by measuring the 
angles, because measures of length can be taken with more accuracy from 
a scale, and transferred to the plan with more exactness than angles can 
be pricked off from a protractor ; but for ordinary surveys, the triangula- 
tion is most frequently and expeditiously plotted by the means of a pro- 
tractor. 

The outlines of the survey having been balanced and plotted in, and 
the subsidiary points, as established by offsets and by triangles, the filling 
in of the interior detail is done by copying from the field book the natural 
features of the ground, in their appropriate position, and according to the 
conventional signs already described. 

In many surveys, as of roads, rivers, canals, and boundaries, the plot 
to be made is but a single line, with a few of the nearest local objects on 
either side. In some instances the angles at each intersection are taken 
merely with reference to the two lines forming the angle, and are therefore 
to be plotted as shown in fig. 23, by laying the protractor at each inter- 
section ; but in other instances, by the method of surveying, the directions 
of all lines are referred to the first as meridian, or if the survey is exten- 
sive, to some number of lines, and the plotting is then expeditiously per- 
formed as in fig. 27. In this system of surveying, instead of fixing the 
vernier at zero, for every back angle the preceding forward angle is re- 
tained except for those lines intended as meridians. 

Surveys for railways, like tliose above, are of lines extensive in lengtli 
but of very little width. In the surveys of preliminary or trial lines, the 
curves at the intersection of lines are seldom introduced ; and in plotting 
it is but the usual method of plotting surveyed lines, by either of the 
methods, according as the survey nuiy have been conducted, with tlie the- 
odolite, or witli the coni})ass. 



TOPOGRAPHICAL DRAWING. 



In plotting curves on a line of location, lay off from the intersection 
of tangents, as C, fig. 32, the distance of the tangent points A and B, and 
find the centre O of the curve, by the erection of perpendiculars to these 




Degree. 


Radii, ft. 


1 
Central Ordinate. 


10 


5729.65 


0.218 


20 


2S64.93 


0.436 


30 


1910.08 


0.655 


40 


14^32.69 


0.873 


50 


1146.28 


1.091 


60 


955.37 


1.309 


70 


819.03 


1.528 


80 


716.78 


1.746 


90 


6:37.27 


1.965 


lOo 


57:3.69 


2.183 



two points, or if the ra- 
dius of the curve is 
known, by describing 
arcs with this radius 
from the same points. 
Eailway curves are de- 
signated by degrees or 
a curve of 1°, 2°, 3°, 
according to the angle of deflection made by two chords of 100 feet 
each. 

Two curves often succeed each other having a common tangent, at the 
point of junction, ilf the curves lie on opj)osite sides of the common tan- 
gent, they form a reversed curve, A B C, (fig. 33,) and their radii may be 

the same or different. 

•^'°* ^^- X I If they lie on the same 

' side of the common 

tangent and have dif- 
ferent radii, they form 
a compound curve, A 
BD. When the radii of 
curvature are known, 
the determination of 
the centres is ob- 
tained easily, by de- 
scribing arcs with the established radii from the tangent points. 

If the radii of curvatures are not known, and it is required to plot 
a compound or reversed curve which shall be tangent at the points A 
and D or C to other straight lines, the change of curvature taking 




364 



TOPOGRAPHICAL DRAWING. 



place at B, then at A erect a perpendicular to the given line, find 
some point on this perpendicular which is equally distant from A and B, 
and this point will be the centre of the curve A B ; through this point 
and B draw an indefinite line, its intersection by the perpendicular to the 
tangent at D will be the centre for the other portion of the compound 
curve ; and its intersection by the perpendicular to the tangent at C will 
give the centre for the reversed. The centres of curves tangent to each 
other must lie in a straight line, passing through their point of con- 
nection. 

When the curves are larger than can be described by the dividers or 
beam compasses, they can be plotted as shown in geometrical problems, 
or points of a curve may be obtained by calculation of their ordinates, 
and the curves drawn from point to point by sweeps and variable curves. 
Approximately, knowing the central ordinate of the curve between two 
points, the central ordinate of one half that curve will be one quarter of 
the first. Thus, fig 32, G D is about one quarter of E F, hence by subdi- 
visions as many points as are necessary may be obtained ; but it should 
be observed, that the greater the number of degrees in the arc, the less 
near to the truth the rule. 



.V.43FcdFallj)iT MUe 



Level 




Fig. 34. 




Fig. 35. 

Fig. 35 represents a plot of a railway line ; in this plot the curve is 
represented as a straight line, the radius of curvature being written in. 
This method is sometimes adopted when it is desirable to confine the plot 
within a limited space upon the sheet, and it is convenient when plotted 
thus directly beneath the profile or longitudinal section (fig. 34). 

In plotting the section, a horizontal or base line is drawn on which are 
laid off the stations or distances at which levels liave been taken ; at these 
points perpendiculars or ordinates arc erected, and upon them are marked 



TOPOGRAPHICAL DRAWING. 



365 



the heights of tlie ground above the base, and the marks are joined by 
straight lines. To express rock in a cut, it is generally represented by di- 
agonal lines ; rivers are represented in section by cross lines or colored in 
blue ; the depth of the sounding in a mud bottom by masses of dots. 

Since it would be in general impossible to express the variations of the 
surface of the ground in the same scale as that adopted for the plan, it is 
usual therefore to make the vertical scale larger than that of the horizontal 
lines one, in proportion of 10 or 20 to 1. Thus, if the horizontal scale of 
the plan be -iOO feet to the inch, the vertical scale would be 40 or 20 feet 
to the inch. 

For the purpose of facilitating the plotting of profiles, profile paper is 
prepared, on which are printed horizontal and vertical lines ; the horizon- 
tal lines being ruled at a distance of ^\ of an inch from each other, every 
fifth line being coarser, and every twenty-fifth still heavier than the others. 
Each of the spaces is usually considered one foot. The vertical lines are 
one quarter of an inch distant from each other, every tenth line being 
made more prominent than the others ; these sj)aces in general represent 
a distance of 100 feet, the usual distance between stations on a railroad. 
Much time is saved by the use of this j)aper, both in plotting, and in read- 
ing the measurements after they are plotted. 

In the plotting of sections across the line, which are extended but little 
beyond the line of the cut or embankment, equal vertical and horizon- 
tal scales are adopted ; these plots are mostly to determine the jDOsition of 
the slope, or to assist in calculating the excavation. To facilitate these, 
cross section paper is prepared, ruled with vertical and horizontal lines, 
forming squares of j\ of an inch each. Every fifth line in each direction 
is made prominent. When cross sections are extended to sliow the grade of 
cross road, or changes of level at considerable distance from the line of 
rail, the same scales vertical and horizontal are adopted as in the longitu- 
dinal section or profile. 

It will be observed in fig. 34, that the upper or heavy line represents 
the line of the rail, the grades being written above ; this is the more usual 
way, but sometimes, as in fig. 36, the profile and plan are combined ; that 




is, the heights and depths above and below the grade line of the road are 



366 TOPOGRAPHICAL DRAWING. 

transferred to the plan, and referred to tlie line in plan, which becomes 
thus a representation both in plan and elevation. 

Cross sections, for grades of cross roads, etc., are usually plotted be- 
neath or above the profile ; they may, if necessary, be plotted across the 
line when plan and profile are combined. 

Besides the complete plans as above, giving the details of the location, 
land plans, so called, are required, showing the position and direction of 
all lines, of fences and boundaries of estates, with but very few of the 
topographical features. The centre line of road is represented in bold 
line, and at each side, often in red, are represented the boundaries re- 
quired for the purposes of way. In general, a width of five rods is the 
amount of land set off, lines parallel to the central line being at a distance 
of two and one half rods on each side ; but when, owing to the depth of the 
cut or embankment, the slopes run out beyond this limit, the extent is de- 
termined by plotting a cross section and transferring the distances thus 
found to the plan, and enclosing all such points somewhat within the 
limits as set off for railway purposes. These plans are generally filed in 
the register's office for the county through which the line passes. 

For Eailway plans prepared for the English Parliament certain regulations are defined. 

The plan must be upon a scale of at least four inches to a mile, and must describe the 
line or situation of the whole of the proposed work, and the lands through which the 
same will be made, and also any communication to be made with the proposed work. If 
the plan is on a scale less than i of an inch to every 100 feet, there must be an additional 
plan upon that scale (1 : 4,800) of any building, yard, and of any ground cultivated as a 
garden. 

The plan to exhibit thereon the distances in miles and furlongs, from one of the ter- 
mini, with memoranda of the radii of all curves less than one mile in length, noted on the 
plan in chains where the curve occurs. "When a tunnel is intended to be constructed, it 
must be marked in by a dotted line on the plan. 

Each distinct property, divided by any visible boundary from another property, should 
have a separate number ; with this exception, that any collection of buildings and grounds 
within the curtilage of a building, belonging to one person and in one occupation, may be 
described under one number ; thus, farm house, yard, &c. When it is necessary to inter- 
pose a number, a duplicate number should be added ; thus, 8«, da. The numbering should 
recommence in every parish. 

All lands included within the limit of deviation, shown by lines drawn on plan, and 
all lands which those lines touch, must be numbered and described. Public roads, and 
private roads if fenced out, should have a separate number. Navigable and mill streams 
must be separately numbered. 

It is sometimes usual, at the completion of a railway, to make plans of 
the works as finished ; and, if a profile of the line, to represent the differ- 



TOPOOKAPIIICAL DRAWING. 



3G7 



ent strata or rocks in the cuts, with their dip or inclination. This is more 
properly a geological profile ; the different rocks are usually distinguished 
by different colors and explained by marginal notes and squares, some- 
times by marks, dots, and cross hatchings, as fig. 37, often by color in ad- 



^<V^i 




dition. Figs I., II., III., lY., represent the primary, secondary, tertiary, 
and recent plutonic rocks. Fig. 1 represents the primary fossiliferous 
strata ; figs. 2, 3, 4, the secondary, tertiary, and recent strata. In the plots 
of geological sections, it is especially requisite that the different strata 
should be accurately represented. 

In plotting hydrometrical or marine surveys, the depths of soundings 
are not expressed by sections, but by figures written on the plan, express- 
ing the sounding or depth below a datum line, generally that of high 
water. The low water line is generally represented by a single continued 
line. The soundings are generally expressed in fathoms, sometimes in feet. 

It is usual, in plotting from a field book, to make first but a rough 
draft, and then make a finished copy on another sheet. In the first, many 
lines of construction, balances of survey, and trial lines are drawn which 
are unnecessary in the copy; outlines of natural features are sketched 
roughly, but the plotting of surveys, and such lines and points as are to 
be preserved in the copy, must be plotted with accuracy. 

The most common way of transferring, for a fair copy, is by superpo- 
sition of the plan above the sheet intended for the copy, and pricking 
through every intersection of lines on the plan and all such points as may 
be necessary to preserve. The clean paper should be laid and fastened 
smoothly on the drawing board, the rough draft should be laid on smoothly 
and retained in its position by weights, glue, or tacks. Tlie needle must 
be held perpendicular to the surface of the plan and pressed through both 
sheets ; begin at one side and work with system, so as not to prick through 
each point but once, nor omit any ; make the important points a trifle the 



368 TOPOGRAPHICAL DRAWING. 

largest. For the irregular curves, as of rivers, make frequent points, but 
veiy small ones. On removing the plan select the important points, those 
defining leading lines ; draw in these, and the other points will be easily 
recognized from their relative position to these lines. When any point 
has not been pricked through, its place may be determined by taking any 
two established points adjacent to the one required, and with radii equal to 
their distance, on the plan, from the point required, describing arcs, on 
the copy, on the same side of the two points, their intersection will be the 
point desired. In this way, as in a trigonometrical survey, having estab- 
lished the two extremes of a base, a whole plan may be copied. For this 
purpose the triangular compasses (p. 27) will be found very convenient. 
In extensive drawings it is very common to prick oif but a few of the 
salient points, and fill in by intersections, as above, or by copying detached 
portions on tracing paper, and transferring them to the copy ; the position 
of each sketch being determined by the points pricked ofi*, the transfer is 
made by pricking through as above, or by transfer paper placed between 
the tracing and the copy. 

Tracing paper is a thin, transparent paper, prepared expressly for the 
purpose of making copies of drawings. Placed above the drawing, every 
line shows through, and is traced directly with the pen, in India ink. 
These tracings are used mostly to preserve duplicates of finished drawings. 
As tracing paper is of too slight a texture to bear much handling, cotton 
cloth is prepared and sold under the name of " vellum tracing paper." 
When it is necessary to use tracing paper drawings to work by, it is usual 
to attach them to sheets of white paper, which serves both to bring out 
the lines and to strengthen the paper. 

Duplicates of drawings are now very neatly executed, and of course 
accurately copied, by the Photographic ]Drocess, but it is more applied to 
mechanical and architectural drawings than topographical. 

An accurate and rapid way of tracing, on drawing paper, plans of small 
extent, is by means of an instrument called a copying glass. It consists 
of a large piece of plate glass set in a frame of wood, which can be in- 
clined at any angle in tlic same manner as a reading or music desk. On 
this glass is first laid the original plan, and above, the fair sheet, and the 
frame being raised to a suitable angle, a strong light is thrown by reflect- 
ors or otherwise on the under side of the glass, whereby every line in the 
original plan is seen distinctly through the fair sheet, and the copy is made 
at once, in ink, as on tracing paper, and finished Avhile being traced. This 
same process, on a small scale, is adopted by putting the plans upon a 
pane of glass in a window. 



TOPOGEAPIIICAL DRAWING. obV 

Plans of great extent cannot be conveniently copied by means of the 
copying glass. Moreover, being often mounted on cloth, which renders 
them opaque, they do not admit of being traced in this way. In such 
cases the copy may be made by means of transfer paper. The plan is first 
traced in ink on tracing paper or cloth, black leaded or transfer paper is 
then placed on the fair sheet, and the tracing paper copy is placed above. 
All is steadied by numerous weights along the edges, or by drawing pins 
fixed into the drawing board. A fine and smooth point is then passed 
over each boundary or mark on the tracing with a pressure of the hand 
sufiicient to cause a clear, pencilled mark to be left on the fair sheet by 
the black leaded or transfer paper. The whole outline is thus obtained, 
and afterwards drawn in ink. The copyist should be careful in his manipu- 
lations, so as not to transfer any other lines than those required, nor leave 
smutches on the fair sheet. 

Plans may be copied on a reduced or enlarged scale by means of the 
pentagraph, but the more usual way is by means of squares. Draw on 
the plan to be reduced, a series of parallel and equi-distant lines, with 
others perpendicular to them at similar intervals, thus covering the whole 
surface with equal squares. On the clean sheet draw a similar set of 
squares, but with their sides to the desired reduced scale ; one-half, one- 
third, (fee, as the case may be. Then copy into each small square all the 
points and lines in the corresponding square on the plan, in their true po- 
sition relative to the sides and corners of the square, reducing each 
distance, by proportional dividers, or by eye as may be necessary, in the 
given ratio. In reducing by the camera lucida, squares on the j)lan are 
brought apparently to coincide w^ith squares in the copy, and the details 
as seen through the prism of the instrument, are then filled in with the 
pencil. This instrument is used in the U. S. Coast Survey office, but it 
cannot reduce smaller than one-fourth without losing distinctness, and it is 
very trjang to the eyes. 

Finishing the Plan or 3fap. — In general, in toj^ographical as in ar- 
chitectural and mechanical drawings, the light is supposed to fall upon the 
surface in a diagonal direction from the upper, left-hand corner. This rule 
is not uniform : by some draughtsmen the light is introduced at the lower 
left, and hills are mostly represented under a vertical light, although the 
oblique adds more to the picturesque eifect. The plan is usually so drawn 
that the top may represent the north, and the upper left-hand corner is 
then the north-west. 

In inking in, commence first with the light lines, since a mistake in 

these lines may be covered by the shade lines. Describe all curves which 
24 



370 TOPOGRAPHICAL DRAWING. 

are to be drawn with compasses or sweeps before the straight lines, for it 
is easier to join neatly a straight line to a curve than the opposite. Ink 
in with system, commencing say at the top ; ink in all light lines rnnning 
easterly and westerly, then all light lines running northerly and southerly, 
then commence in the same way and draw in the shade lines. It wdll of 
course be understood that elevated objects have their southern and eastern 
outline shaded, whilst depressions have the northern and western ; thus 
in conventional signs roads are shaded the opposite to canals. Having 
inked in all lines that are drawn with a ruler or described with compasses, 
commence again at one corner to fill in the detail, keeping all the rest of 
the plan except what you are actually at work upon covered with paper, 
to protect it from being soiled. The curved lines of brooks, fences, &c., 
are sometimes drawn with a drawing pen, sometimes with a steel pen or 
goose quill. The latter are generally used in drawing the vertical lines 
of hills. 

Boundary lines of private properties, of townships, of counties, of 
states, &c., are indicated by various combinations of short lines, dots and 



crosses, thus : 



All plans should have meridian lines drawn on them, also scales. Plate 
XCI. shows some designs for meridians. In these diagrams it will be 
observed that both true and magnetic meridians are drawn ; this is desira- 
ble when the variation is known, but in many surveys merely the magnet- 
ic meridian is taken ; in these cases this line is simply represented with 
half of the barb of the arrow at the north point, and on the opposite side 
of the line from the true meridan. Scales are drawn or represented in va- 
rious forms, as may be seen in the following plates, or the proportion of 
the plan to the ground is expressed decimally, as the number of feet, 
chains, etc., to the inch. 

Lettering. — ^The style in which this is done very much aifects the gen- 
eral appearance of the plan. Great care must be taken in the selection 
and character of the type, and in the execution. The usual letters are the 



K O ]\I A N . 



ABCDEFGHIJKLMNO 
PQRSTUYWXYZd: 



PLATE XCr. 




TOPOGRAPHICAL DKAWING. 371 



MALL ROMAN, 



abcdefghijklmnopqrstuv 

wxyz,;:. 

1234567890 



ITALIC. 



ABCDEFGHIJKL 

MJYOPQRSTUFW 

XYZ^ 

ahcdefghijklmnopqrst 
uvwxyz;: 



GOTHIC, OR EGYPTIAN 



ABCDEFCHIJKLMN 
OPQRSTUVWX 

■ mm fSKf H ■ ■ ■ 



3T2 



TOPOGRAPHICAL DRAWING. 



OLD ENGLISH SCRIBE BLACK. 



GERMAN TEXT. 










^ 




i> 
















^ 



«^<^^f^|||iC«iHf^t^^l«^^^'S^5- 



CLARENDON 



ABCDEFGHIJKLMNOP 
aRSTUVWXYZ& 



TOPOGRAPHICAL DRAWING. 



373 



SMALL CLARENDON. 



abcdefghijklmnopqrstu 



vwxyz 






GRECIAN 



HCDEFEHUKLMNOFilRliT 



OLD ENGLISH. 



^ 



<1 m !] 



ii 



ljWiitfl|i|irjSsli!if| 



tm 



> <> -> <^ 






iiiiiFiiiiiLiiiPiiififW 

If 11,11. 



374 TOPOGRAPHICAL DRAWING. 

ENGLISH OUTLINE. 

PL XCII. represents a mechanical metliod of constructing letters and 
figures. This plate should be copied hj the draftsman, and on a much 
larger scale, by drawing first the system of squares or parallelograms, and 
then sketching in the letters ; in this way well formed and proportioned 
letters can always be made, and from a selection of alphabets the lettering 
may be selected and transferred to the plan. 

PL XCIII. are examples of titles, intended merely as an illustration of 
the form of letters and their arrangement, the scale being much smaller 
than that used on plans, except such as are drawn to a very small scale. 
It will be observed that the more important words are made in promi- 
nent type. The lower part of the title should always contain, in small 
character, the name of the party making the survey, and also the name 
of the draftsman, with date of the execution of the plan : if the survey 
was made some time previous, the date of the survey should be given. 
If the plan is compiled from several surveys, the authorities should, if 
possible, be given. The lettering of the title should be in lines parallel to 
the bottom of the plan, and, in general, the great mass of lettering in the 
body of the plan is formed in similar lines ; but curved lines are often not 
only essential, but they materially contribute to the beauty of the plan. 
Thus on crooked boundaries on outlines of maps, the lettering should fol- 
low the general curve of the boundary ; also on crooked rivers, lakes, 
seas, &c. ; on irregular or straggling pieces of land, in order to show the 
extent, connection, or proprietorship thereof, the lettering should follow 
the central line of such a tract ; and if pieces of land be very oblong in 
form but regular in outline, the lettering wdll be central in the direction of 
the longest side. The lettering of roads, streets, &c., is always in tlie di- 
rection of the line of road. Curved lines of lettering are often introduced 
into extended titles to take off the monotonous appearance presented by a 
great number of straight lines of writing. 

The direction of all lettering should be so as to be read from left to 
right. If shades or sliadows arc introduced to give relief or break up the 
monotony, they should be uniform with the rest of the plan. 

On the Spacing of Letters. — It will be observed that letters vary very 
considerably in their width, the / being the narrowest, and the irthc 



PLATE XCII. 



374 







mmmm 



PI.XClJj 



:M4 




PROPOSEl] BAIL ROAD 

BETWEEN 




T^E 



TIAT\L m 





%\\tt'i't^i\> %; \>t<mn Bt3 




:^^ 



Sca2^ o/^ Ck^zl?^s : 



JIfqy, /SS7. 



TOPOGRAPHICAL DRAWING. 



'6iD 



widest : if therefore the letters composing a word be spaced off at equal 
distances from centre to centre, the interval or space between the letters 
will be more in some cases than in others. Thus, in the word 

R A I L W A Y 

To avoid this, write in first one letter, and then space off a proper interval, 
and then write in the next letter, and then space off the interval as before, 
and so on, thus, 

RAILWAY 

When, as frequently happens, the words are very much extended, in order 
to embrace and explain a large extent of surface or boundary, and the 
space occupied by the letter is small in comparison with the interval, the 
disparity of intervals wdll not be noticed, and the letters may be then laid 
off at equal spaces from centre to centre, thus : 



I 



W A 



When the lines of lettering are curved, the same rules for spacing are to 
be observed as above. If the letters are upright, as Homan or Gothic, the 
sides of each letter are to be j)arallel to the radius drawn to the centre of 
the letter, and the bottom and top lines at right angles to it. If the let- 
ters be inclined as Italic letters, then the side lines of the letters must be 
inclined to the central radial line, as on a horizontal line they are inclined 
to the perpendicular. 




In laying off letters by equal intervals, it is usual to count the number 
of letters in the word, and fix the position on the plan of the central one, 
and then space off on each side : this is particularly important in titles, 
when it is necessary that many lines should have their extremities at uni- 



376 TOPOGEAPHICAL DRAWING. 

form distances from the centre line. In laying off the title, we determine 
what is necessary to be included in the title, the space it must occupy, the 
number of lines necessary, and the style and arrangement of characters to 
be used. Tlius, if the title were, plan of a proposed terminus of the Har- 
lem Railroad at 'New York, 1857, knowing the space to be occupied, we 
can write the title thus : — 



^ 



o^ me 

"We now draw parallel lines at intervals suited to the character of the type 
we intend to employ for the different words. Harlem Railroad is the line 
to be made most prominent ; this, calling the interval betw^een the words 
one letter, includes 15 letters ; or, if we consider Z, with its proper interval, 
but half a letter, (which will be found a very good rule in spacing,) \^\ ; 
hence the centre of the line will be 7^ letters from the beginning, or \ of 
the space occupied by the letter H and its interval. Draw a perpendicu- 
lar line at the centre, and write in R in such a character as may suit the 
position to be filled, and lay off by letters and spaces the other letters. 
The line Harlem Railroad is intended to occupy the whole length of 
space ; that is, it must be the longest line in the title, and the lines above 
and below must gradually diminish, forming a sort of double pyramid. 
Projposed Terminus includes 16| letters, the I and interval between the 
Avords being rated as above, we find the centre to be nearly midway be- 
wccn the words. These words including more letters, and being confined 
within less space, must be in smaller character than the preceding ; and as 
a fiirtlicr distinction, a different style should be adopted. Having deter- 
mined tliis, we proceed to write in tlie letters as before, and in the same 
way witli the otlier lines, the prepositions as unimportant are always writ- 
ten in small type. 



TOPOGRAPHICAL DRAWING. 



377 




X)f the. 



of the . 

HARLEM RAILROAD 

at 



EW YORK 

1857 



In general it is better that letters slionld be first written on a piece of 
paper, distinct from tlie plan, as repeated trials may be necessary before one 
is arranged to snit the draughtsman. Having formed a model title, it may be 
copied in the plan by measures or by tracing and transfer paper. Tliere 
are some w^ords, such as Plan, Map, Section, Scale, Elevation, &c., which, as 
they are of constant occurrence^ may be cut in stencil ; sometimes whole 
alphabets are thus cut and words compounded. It will be found very con- 
venient for a draughtsman if he makes tracing or copies of such titles as 
he meets with, and preserves them as models ; for there is no manipulation 
on a plan that contributes more to the effect than good lettering and arrange- 
ment of titles, and considerable practice should be expended in acquiring 
a facility in lettering, and for the first start, perhaps nothing w^ll be found 
more valuable than tracing good examples. 

We have treated of mechanical methods by which most j)Grsons can 
learn to form letters and words ; but it must be borne in mind that the 
distances between letters on the plan are only intended to suit the eye ; 
if therefore a person accustom himself to spacing, so that his eye is cor- 
rect, tliere will be no necessity of laying off by dividers ; in this mode, 
such letters as A and Y, L and T are brought nearer each other than tlie 
regular interval. In general it may be observed in reference to to the 
lettering of Topographical Drawings, stiff letters like those of stencil 
should not be introduced, but there should be such variety, incident on 
construction by the pen, as may be consonant with the rest of the drawing. 



I 



378 TOPOGRAPHICAL DRAWING. 



TINTED TOPOGRAPHICAL DRAWING. 

We have hitherto treated of the representation of the features of the 
country by the pen only, but it may be done full as effectively and much 
more expeditiously by means of the brush and water colors, either by 
India ink alone, or by various tints, or by the union of both. 

The most important colors for conventional tints are, (besides India 
ink). Indigo (blue), Carmine (or crimson lake), and Gamboge (yellow), 
used separately or compounded. Besides these. Burnt Sienna, Yellow 
Ochre, and Yermilion are sometimes used, although the three first are 
susceptible of the best combinations, and the othei^ are generally used 
alone. 

The following conventional colors are used by the French Military En- 
gineers in their colored topography. Woods, yellow / using gamboge and 
a very little indigo. 'Grass land, ^r^e?^/ made of gamboge and indigo. 
Cultivated land, Irown ; lake, gamboge, and a little India ink; "Burnt 
Sienna" will answer. Adjoining fields should be slightly varied in tint. 
Sometimes furrows are indicated by strips of various colors. Gardens are 
represented by small rectangular patches of brighter green and 'brown. 
Uncultivated land, marbled green and light brown. Brush, brambles, &c., 
marbled green and yellow. Heath, furze, &c., marbled green and jphik. 
Vineyards, purple ; lake and indigo. Sands, a light brown ; gamboge 
and lake ; " Yellow Ochre " will do. Lakes and rivers, light blue, with a 
darker tint on their upper and left hand sides. Seas, dark blue, with a lit- 
tle yellow added. Marshes, the blue of water, with spots of grass green, 
the touches all lying horizontally. Boads, brown ^ between the tints for 
sand and cultivated ground, with more India ink. Hills, greenish brown ; 
gamboge, indigo, lake and India ink. Woods may be finished up by 
drawing the trees and coloring them green, with touches of gamboge to- 
wards the light, (the upper and left hand side,) and of indigo on the op- 
posite side. 

In addition to the conventional colors, a sort of imitation of the con- 
ventional signs already explained are introduced in color with the brush, 
and shadows are almost invariably introduced. Tlic light is supposed to 
come from the upper left liand corner, and to foil nearly vortical, but suf- 
ficiently oblique to allow of a decided light and shade to the slopes of 
hills, trees, c^^c. The shadow of any ol)ject will therefore surround its 



TOPOGRAPHICAL DRAWING. 370 

lower right hand outlines. After the shadow has been painted, the out- 
line of the object is strengthened by a heavy black line on the side oppo- 
site the light. Tlie flat tmts are first laid on as above, and then the con- 
ventional signs are drawn in with a pencil and colored in with appropriate 
and more intense tints ; the shadows are generally represented in India ink. 

Hills are shaded, not as they would aj^pear in nature, but on the con- 
ventional system of making the slopes darker in proportion to their steep- 
ness : the summit of the hiojhcst rans-es beino: left white. 'Eiis arranofc- 
ment, though obviously incorrect in theory, has the advantage of being 
generally understood by those not accustomed to plan drawing, and is 
also easy of execution. Wash the surface first with the j^roper flat 
tint, trace in with a pencil, outlines ; then lay on in India ink tints pro- 
portioned in intensity to the height of the hills and steepness of the 
slopes. To soften the tints two brushes are used, one as a color brush, the 
other as a water brush : the tints are laid on with the first, and softened 
by passing the water brush rapidly along the edges. The water brush 
must not have too much water, as it would in that case, lighten the tint 
to a greater extent than is intended, and leave a ragged harsh edge. Tints 
may be applied in very light shades, one tint over another, with the 
boundary of the upper tint not reaching the extreme limit of the tint 
below it. When depth of shade is required, it is best produced by appli- 
cation of several light tints in succession : no tint is to be laid over the 
other until the first is drv, and a little indiiro mixed with the India ink im- 
proves its color and adds to the richness of efi'ect. 

When woods have to be represented, the shading used for the trees 
instead of interfering with the shadows due to the slopes, may be made to 
harmonize with them.^ and contribute to the general efi'ect by presenting 
greater or less depth, according to the position of the woods on the sides, 
or summits of the hills. 

An expeditious and efi'ective way of representing hills with brush, a 
species of imitation of hills drawn with a pen on the vertical system, is 
efi'ected by pressing out flat the brush to a sort of comb-like edge ; draw- 
ing this over a nearly dry surface of India ink, and then brusliing lightly 
or more heavily between the contours, according to the steepness of the 
slope, each of the comb-like teeth making its mark. 

Rivers and masses of water may be shaded in with a color and water 
brush as above, or by superposition of light tints, a shadow may be 
thrown from the bank towards the light, and the outline of this bank 
strengthened with a heavy black line. The tints are to be in indigo, the 
shadows in India ink. 



380 TOPOGRAPHICAL DRAWING. 

Topograpliical drawings may be made in water color with but one tint, as 
India ink, or ink mixed with a little sepia. The conventional signs are in 
imitation of pen drawings, the hills in softened tint, or drawn with the 
comb-edged brush, and the rivers shaded with superposed tints. 

Most artistic and effective drawings are made of hills as they would 
appear in nature, under an oblique light : the sides of the hills next the 
light receiving it more or less brilliantly, according as they are inclined 
more or less at right angles with its rays, and the shades on the sides re- 
moved from the light increasing in intensity as the slopes increase in steep- 
ness. This style may be rendered most expressive by a skilful draughts- 
man, especially when the character and strike of the hills are favorable to 
the direction of the light, but with this style of representation the hills are 
generally made to partake more or less of the same character, appearing 
almost uniformly steepest on the sides removed from the light. It partakes 
therefore more of the artistic character, more difficult to execute, and 
conveying information in a more vague manner than by the common to- 
pographical conventionalities. As a picture it must be left to explain 
itself; the writing and outline necessarily introduced on the plan con- 
tribute to mar its effect. 

In preparing the paper for a tinted drawing it must be damp-stretched 
upon the drawing board in such a manner that the moisture of the color 
will not cause undulations or blisters on the surface : this process is pre- 
viously described at page 37. Having prepared a sheet of paper accord- 
ing to the directions there given, first draw in the lines in pencil, and af- 
terwards repeat them with a very light ink line : a soft sponge well satu- 
rated should then be passed quickly over the surface of the drawing, in 
order to remove any portions of the ink which would be liable to mix with 
the tint and mar its uniformity. When the paper is dry proceed to lay on 
the conventional tints. 

Great care is necessary in preparing and combining the different colors, 
and attention to certain mechanical conditions and rules must be observed 
in order to insure neatness and despatch in execution. The cakes of color 
are quite brittle, and it is well to moisten the end and allow it to soften 
slightly before using, then rub upon a perfectly clean palette, with a few 
drops of pure water, a sufficient quantity of color to tinge to the proper 
intensity as much water as will be required for the whole drawing, This 
should be thoroughly mixed with the brush, and as often as the brush is 
filled, to insure uniformity in the tint. 

Previous to applying tha tint, it is well to moisten the surface to be col- 
ored with clean water, which will prevent the tint from drying too rapidly 



TOPOGRArUICAL DRAWING. 381 

at tliG edges. In tinting never allow tlie edge to dry until the -whole sur- 
face is covered : leave a little superfluous color along the edge wliilst 
filling the brush. Great caution is necessary in approaching the outlines 
of the drawing, and the point of the brush should be used so as not to 
overrun the lines. 

In applying a flat tint to large surfaces, let the drawing board be in- 
clined upwards at an angle of 5 or G degrees, so as to allow the color to 
flow downwards over the surface. With a moderately full brush com- 
mence at the upper outline, and carry the color along uniformly from left 
to right and from right to left in horizontal bands, taking care not to over- 
run the outlines, in approaching which the point of the brush should be 
used, and at the lower outline let there be only sufficient color in the 
brush to complete the tinting. 

"No color should be allowed to accumulate in inequalities of the paper, 
but should be evenly distributed over the whole surface. 

Too much care cannot be given to the first application of color ; as any 
attempt to remedy a defect by washing or applying fresh tints will be 
found extremely difficult, and to generally make bad worse. 

Erasers should never be used on a tinted drawing to remove stains or 
patches, as the paper when scratched, receives the tint more readily, and re- 
tains a larger portion of color than other parts, thereby causing a darker tint. 

Marbling is done by using two separate tints, and blending them at 
their edges. A separate brush is required for each tint ; before the edge 
of the first is dry, pass the second tint along the edge, blending one tint 
into the other, and continue with each tint alternately. 

In reference to the general efi'ect to be produced in tinted topographi- 
cal drawings, as to intensity, every thing should be subordinate to clear- 
ness, no tint should be prominent or obtrusive. Tints that are of small 
extent must be a little more intense than large surfaces^ or they will appear 
lighter in shade. Keep a general tone throughout the whole drawing. 
Beginners will find it best to keep rather low in tone, strengthening their 
tints as they acquire boldness of touch. 

In lettering tinted drawings, let the letters harmonize with the rest of 
the plan ; let them be in tint more intense than the topography, prominent 
but not obtrusive. 

Flourishes around the titles may be used on handsome estate maps, and 
on engraved maps of countries. They should be used in proportion to the 
degree of finish bestowed on the rest of the map ; and while they give 
grace and elegance to the title when used in moderation, care should be 
taken to prevent their having too prominent an appearance. 



382 TOPOGRAPHICAL DKAWmG. 

Plate XCiy. is a map of the Harbor and City of New Haven, re- 
duced from the charts of the U. S. Coast Survey, without the depth of 
soundings or the marks of shoals. 

Plates XCY. and XCYI. are examples of topographical drawings, the 
one in ink and the other in color. 

Plate XCYII. is a geological map from " Blake's Geological Survey of 
California." On geological maps sections are similarly represented, and 
plans are colored in patches according to the formation. Shades of India 
ink usually represent coal measures ; of blue, limestone ; of pink, the igne- 
ous rocks, as trap, granite, &c. In all cases, there are small blocks of 
color at the margin of the map, to designate the mineral represented by 
each. 

In all the departments of drawing, we have thus tried to illustrate as 
fully as the limits of this work would admit, the general principles of rep- 
resentation and the rules of projection; but in addition, we would recom- 
mend to every one who wishes to make himself a perfect draughtsman, 
that he should collect good charts and drawings, study them, and in his 
leisure moments copy them. In this way he will acquire a readiness of 
manipulation, and ease and freedom of expression. 



PLATE XCIY. 



382 




PL.XCV. 



382. 







PLATx^Xai 



382 




ts; 














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Arr, r'-.olo- l.iidoqraar.it ,'.0 N T. (Ojtcrn«j'^roc:Sj; 



'LAT'E XC'TIl. 



GEOLOGICAL MAP 
(•r ;» |jail ()(' tlif Sfsa*' .1,' 

h TAliiFORMA 

hxptoreil in 18i') by 

^"•^N LIKIT H S WILLIAMSON V S TOP tNT." 




GTIE A T B A "^1% 



a. wutfJius 



Hume* i>. to. L .\ I ^ 1^^ i^ 



^ -JffrnJjX.,^ 



I 






Graitiiic 8t-Xetajiiorpluc 



Erupted Granite ft Syenite 



; Serpeatuie Trap CrL-epu.stane BrPoi-pliyry ^ samiLBa, 

SAX 

3 Basaltic La^a Found in ti taiU Land or plateau at 

Fort Miller anitU many ptint* /urther north net rrprtsfnttd trn the map} 



^ ifrtaitiorphu' Slates cliiefh- (lu\ Slate ft-Ialcose Slate Aun/erons.) 




, lIThite fttcTA'^stsTlhie Liuiewtoite Mrtairuirpfwsi-d 




Am Tholo-LithogTaphic,Co.M.Y. (OsbBrn«»F'rocs3<' 



^LATE XC^/Il 




IB (kardtic SrMetanxDrpTiic 

■jj} Erupted Gramte * Syenite 

r~i Serpentine Trap Greeiis tone PfPorpltyiry - 

~~1 Basaltic Lava Found in a tahle Land ot Plnlemi at 

I FoH Miller and tU many pbirdg farther north not rrprteented On themap) 

I 

I 

^ ifetaiitorphu' Slates cliiefh- Clav SJateft-Ialcose SLite Aunferoiis.) 

J ^Vlntfe St crv'^staTliie taiiie«toii-e Mcta/rwrp/w 



Am.rholo-LilhogTaphic.Co.M.Y.fOjbornesFn 



PHOTOGRiVPIlY. 383 



COPYING OF DRAWINGS BY PirOTOGRxVPIIY. 

As it is now quite common to make copies of drawings by Photogra- 
pliY, it seems not out of place to giYc a brief abstract of the process as 
taken from the " Manual of Photography " by Robert Hunt. 

The most simple method of obtaining sun pictures is that of placing the drawing to be 
copied on a sheet of prepared paper, pressing it close by a piece of glass, and exposing the 
arrangement to sunshine ; all the parts exposed darken, while those covered are protected 
from change, the resulting picture being white on a dark ground: such pictures are called 
negative pJiotograpJis. Positive photographs, those which are correct copies of the draw- 
ing, are obtained by the super-position of the negative upon another piece of prepared pa- 
per. Some kind of copying frame is an indispensable requisite to the photographer ; it is 
used for copying all objects by transmission, and for multiplying the original pictures ob- 
tained by means of the camera obscura from nature. Some prefer two plates of stout 
plate glass pressed very closely together with clamps and screws ; but as the intention is 
to bring the object to be copied and the sensitive paper into the closest possible contact, 
numerous mechanical contrivances will suggest themselves for the purpose to the ingenious. 

Select a paper of a uniform texture, free from spots and of equal transparency, choos- 
ing the oldest rather than the newest varieties. In the preparation of paper, the only ap- 
paratus required are some very soft sponge brushes and large camel-hair pencils, (no metal 
should be employed in mounting the brushes, as it decomposes the silver salts,) a wide 
shallow vessel capable of receiving the sheet without folds, a few smooth planed boards 
sufficiently large to stretch the paper upon, and a porcelain or glass slab ; also some good 
wJiite blotting paper, soft linen or cotton cloth, a box of pins, a glass rod or two, some por- 
celain capsules, and some beaker glasses, graduated measures, scales and weights. 

1 would advise the amateur to start upon his studies with but three solutions: 1st, 
chloride of sodium ; 2d, nitrate of silver ; 3d, hyposulphite of soda. 

Muriated Papers, as they are termed, are formed by producing a chloride of silver on 
their surface, by washing the paper with the solution of the chloride of sodium, {muriate 
of soda,) or any other chloride, and when the paper is dry, with a solution of nitrate of 
silver. Take some flat deal boards perfectly clean, pin upon them by their four corners, 
the paper to be prepared, observing the two sides of the paper, and selecting that side to 
receive the preparation which presents the hardest and most uniform surface. Then, dip- 
ping one of the sponge brushes into the solution of chloride of sodium, a sufficient 
quantity is taken up by it to moisten the surface of the paper without any hard rubbing, 
and this is to be applied with great regularity. The papers being "salted," are allowed to 
dry : a great number of these may be prepared at a time, and kept in a portfolio for use. 
To render these sensitive, the papers being pinned on the boards or carefully laid upon 
folds of white blotting paper, are to be washed over with the nitrate of silver, applied by 
means of a camel-hair pencil, observing the instructions previously given as to the method 
of moving the brush upon the paper. After the first wash is applied, the paper is to be 
dried, and then subjected to a second application of the silver solution. Thus prepared, 
it will be sufficiently sensitive for all purposes of copying by application. The second 
wash is applied for the purpose of insuring an excess of the nitrate of silver in combina- 



384 PHOTOGEAPHY. 

tion, or more properly speaking, mixed with tlie chloride. The proportions in which the 
chloride of sodium have been used are exceedingly various ; in general, the solutions have 
been made too strong, but several chemists have recommended washes that are as much 
too weak. For copies of engravings or drawings the following proportions should be used : 

Chloride of sodium, twenty-five grains to an ounce of water. Mtrate of silver ninety- 
nine grains to an ounce of distilled water. 

The paper is first soaked in the saline solution, and after being carefully pressed between 
folds of blotting-paper and dried, it is to be washed twice with the solution of silver, dry- 
ing it by a warm fire between each washing. 

Of all the fixing agents, the hyposulphite of soda is decidedly the best. 

The jjicture, or as many of them as there may be, is to be soaked in warm water, but 
not warmer than may be borne by the finger ; this water is to be changed once or twice, and 
the pictures are then to be well drained, and either dried all together, or pressed in clean 
and dry blotting-paper to prepare them to imbibe a solution of the hyposulphite of soda, 
which may be made by dissolving an ounce of that salt in a quart of water. Having 
poured a little of the solution into a flat dish, the pictures are to be introduced one by 
one ; daylight will not now injure them ; let them soak for two or three minutes, or even 
longer if strongly printed, turning and moving them occasionally. The remaining unre- 
duced salts of silver are thus thoroughly removed by soaking in water and pressing in 
clean blotting paper alternately ; but if time can be allowed, soaking in water alone will 
have the efiect in twelve or twenty-four hours, according to the thickness of the paper. It 
is essential to the succees of the fixing process, that the paper be in the first place thorough- 
ly penetrated by the hyposulphite, and the sensitive matter dissolved ; and next, that the 
hyposulphite compounds be effectually removed. 

The hyposulphite of silver being formed, it has to be dissolved out of the paper, the 
fibres of which hold it with a strong capillary force ; and it is only by very long continued 
soaking that all can be removed. The slight mechanical aid afforded by dapping the surface 
of the paper with a soft sponge well filled with water, greatly accelerates the removal of 
the salt ; and when the paper ceases to taste sioeet, we may depend upon the permanence 
of the photograph. 



PEESPECTIVE DRAWING. 



3bo 



PEESPECTIYE DEAWII^G. 




TiiE science of Perspective is the representation by geometrical rules, upon 
a plane surface, of objects as tliey appear to the eye, from any point of 
view. 

All the points of the surface of a body 
are visible by means of luminous rays 
proceeding from these points to the eye. 
Thus, let the line A B (fig. 1) be placed 
before the eye, C, the lines drawn from 
the dififerent points 1, 2, 3, 4, &c., repre- 
sent the visual rays emanating from each 
of these points. It is easy to understand 
that, if in the place of a line a plane or rig. i. 

curved surface is substituted, the result will be a cone of rays. 

Let A B (fig. 2) be 
a straight line, and let 
the globe of the eye be 
represented by a cir- 
cle, and its pupil by 
the point C. The ray 
emanating from A, en- 
tering through C, will 
proceed to the retina 
of the eye, and be de- A' 
picted at a. And as 
it follows that all the 
points of A B will send 
rays, entering the eye 
through C, the whole image of A B will be depicted on the retina of \h^ 
25 




Fiff. 2. 



«3-S6 PERSPECTIYE DRAWING. 

eye in a curved line a 3 I. Conceive the line AB moved to a greater 
distance from tlie eye, and placed at A^ B', then the optic angle will be 
reduced, and the image a' 3 V will be less than before ; and as our visual 
sensations are in proportion to the magnitude of the image painted on the 
retina, it may be concluded that the more distant an object is from the eye, 
the smaller the angle under which it is seen becomes, and consequently 
the farther the same object is removed from the eye the less it appears. 

Observation has rendered it evident, that the greatest angle under 
which one or more objects can be distinctly seen, is one of 90°. If be- 
tween the object and the eye there be interposed a transparent plane (such 
as one of glass m n), the intersection of this plane with the visual rays are 
termed perspectives of the points from which the rays emanate. Thus a 
is the perspective of A, h of B, and so on of all the intermediate points ; 
but, as two points determine the length of a straight line, it follows that 
ah iQ the perspective of A B, and a^ V the perspective of A'' B^ 

It is evident from the figure that objects appear more or less great ac- 
cording to the angle under which they are viewed ; and further, that ob- 
jects of unequal size may appear equal if seen under the same angle. 
For draw/"^, and its perspective will be found to be the same as that of 
A^B^ 

It follows also, that a line near the eye may be viewed under an angle 
much greater than a line of greater dimensions but more distant, and 
hence a little object may appear to be much greater than a similar object 
of larger dimensions. Since, therefore, unequally sized objects may ap- 
pear equal in size, and equally sized objects unequal, and since objects are 
not seen as they are in effect, but as they appear under certain conditions, 
perspective may be defined to be a science which aifords the means of rep- 
resenting, on any surface whatever, objects sucli as they appear when seen 
from a given point of view. It is divided into two branches, the one 
called linear perspective, occupying itself with the delineation of the con- 
tours of bodies, the other called aerial perspective, Avitli the gradations of 
colors produced by distance. It is the former of these only, that is pro- 
posed here to be discussed. 

The perspective of objects, tlien, is obtained by tlie intersection of the 
rays which emanate from them to the eye, by a plane or other surface 
(which is called the picture), situated between the eye and the objects. 

From the explanation and definition just given, it is easy to conceive 
tluit linear perspective is in reality the problem of constructing the section, 
l)y a surface of some kind, of a pyramid of rays of which the summit and 
the l)asc are given. The eye is the summit, tlie base may be regarded as 



I 



PERSPECTIVE DRAWING. 



387 



tlie whole visible extent of the object or objects to be represented, and the 
intersecting surface is the picture. 

A good idea of this will be obtained by suj^posing the 2)icture to l)c a 
transparent plane, through which the object may be viewed, and on wliicli 
it may be depicted. 

In addition to the vertical and horizontal planes Avith whicli we are fa- 
miliar in the operations of projection, several auxiliary planes are em- 
ployed in perspective, and particularly the four following : 



T 








iV^ 






F J 


^ 


\d 






,.'-''' 




r..--'' : 


L^ 








i ^ 






D 




\ E 




^.•'' 






/^ 


.^ 


^/ ^^S 



M 



Fi£ 



1. The horizontal plane A B (fig. 3), on which the spectator and the 
objects viewed are supposed to stand, for convenience supposed perfectly 
level, is termed the ground plane. 

2. The plane M 'N, w^hich has been considered as a transparent plane 
placed in front of the spectator, on which the objects are delineated, is 
called the plane of projection or the plane of the picture. The intersec- 
tion M M of the first and second planes is called the line of projection, the 
ground^ or hase line of the picture. 

3. The plane E F passing horizontally through the eye of the spectator, 
and cutting the plane of the picture at right angles, is called the horizontal 
plane, and its intersection at D D with the j^lane of the picture is called the 
horizon line., the horizon of the picture, or simply the horizon. 

4. The plane S T passing vertically through the eye of the spectator, 
and cutting each of the other planes at a right angle, is called the central 
plane. 

Point of view^ or point of sights is the point where the eye is supposed 
to be placed to view the object, as at C, and is the vertex of the 
optic cone. Its projection on the ground plane S is termed the station 
point. 



388 



PERSPECTIVE DRAWING. 



The projection of any point on the ground plane is called the seat of 
that point. 

Centre of mew (commonly, though erroneously, called the point of 
sight), is the point Y where the central vertical line intersects the horizon 
line ; a line drawn from this point to the eye would be in every way per- 
pendicular to the plane of the picture. 

Points of distance^ are points on the horizontal line, as remote from 
the centre of view as the eye. 

Vanishing points^ are points in a picture to which all lines converge 
that in the original object are parallel to each other. 

Parallel persjpective. — An object is said to be seen in parallel perspec- 
tive when one of its sides is parallel to the plane of the picture. 

Angular per sj>ective. — An object is said to be seen in angular perspec- 
tive when none of its sides are parallel to the picture. 

To find the perspective of points, as the points m, s, (fig. 4) in the ground 




Fis. 4. 



■l! 



plane, the same letters designating similar planes and points as in fig. 
3. From the point m draw a line to the point of sight C, and also to the 
station point S, at the intersection of the line m S with the base line M S\ 
erect a perpendicular cutting the line m C, the intersection mJ will be the 
perspective projection of the point w., on the plane of the picture M Y. 
The point s being in the central plane, its projection must be in the in- 
tersection of that plane by the plane of tlie picture, as the point ^9' the 
intersection of the central vertical line by the line s C. The point v 
being botli in the central and liorizontal ])hinc, its projection in the piano 
of the picture must be in the intersection of all three planes, or at tlie 



DRAWING IN PERSPECTIVE. 



389 



point of view Y. The point A being in tlie liorizontal plane, its projec- 
tion must be in the intersection of tliis plane with the plane of the picture, 
or the intersection li' of the horizon line by the line li C. The points li 
and on being in the same vertical line, the points li! and ni' must also be in 
the same vertical line in the plane of the j^icture, and the position of K' 
might be determined by the intersection of A C by the perpendicular to 
the base line at its intersection by m. S. 

Connect the points hvsm,, and also their projected perspective points 
h' Y s' on\ and we find that when an original line is jparallel or jperjpendic- 
ular to the lase of the picture^ the perspective of that line wdll also be par- 
allel or perpendicular to it. 

Fig 5. Draw the diagonals M s and m S', project as in preceding fig- 
ure the points m and s into the plane of the picture, draw M m, M S' and 
S' m' ; now since m and M are the extremities of a line perpendicular to 




Fig. 5. 

the plane of the picture, the line m' M must be the projection of this line 
on the plane of the picture, and if this line be extended it will pass through 
Y, wdiich may be demonstrated of all lines perpendicular to the plane of 
the picture ; hence the perspective direction of lines perpendicular to the 
picture is to the centre of vieiv. 

If the line on' S^ be extended, it will pass through the point D, and if 
M s' be extended it will pass through a point in the line of the horizon at 
a distance from Y equal to Y D ; by construction D Y has been made equal 
to Y C, and as this demonstration is applicable to other similar lines, and 
since M m 5 S' is a square ; hence the perspective direction of all lines. 



390 DRAWING IN PERSPECTIVE. 

Quaking an angle of 4:6'^ with the jplane of the picture, is towards the jgoint 
of distance. 

Having thiis illustrated the rules of parallel perspective, we now pro- 
ceed to apply them to the drawing of a square and cube, PI. XCIX. The 
same letters are employed in similar position as in preceding figures. 

It is necessary to premise that the student should draw these examples 
at least three times the size of those in the plate. 

Let A and B (fig 1,) represent the plan, or situation upon the ground, 
of two squares, of which a perspective representation is required. First 
draw the line M M, which represents the base line of the picture ; make S 
the station point or place of the observer, and draw lines or rays from all 
visible angles of the squares, to S ; then draw the lines S M, parallel to the 
diagonal lines of the squares. E'ow draw M^ M' parallel to M represent- 
ing the base line of the picture in elevation ; then draw S^ Y, the vertical 
line immediately opposite the eye ; let the distance, S^ Y, be the height of 
the eye from the ground, and draw D D the horizontal line ; Y being the 
centre of view ; let fall perpendicular lines from the angles a and h of the 
plan of the square A, and also from the point <?, w^here the ray from the 
angle e intersects the base line, M M, and from a' and V ^ where a a' and h V 
intersect the base or ground line M^ M^, draw lines to the centre of view, 
Y; and 6^ where the perpendicular line from c intersects the line 5 'Y, will 
give the apparent or perspective width of the side h e ; from e' draw a 
line parallel to a' 1)\ and the perspective representation of the near- 
est square A, is complete. In order to prove the accuracy of this per- 
formance, it is necessary to try if the diagonal lines, a' e\ and 1)' f\ incline 
respectively to the points of distance, D D, on the horizontal line : if so, it 
is correct. The square B is drawn in precisely the^ same manner, and w^ill 
be easily understood by observing the exam2)le. 

The plans of the two cubes C and D, are the same as the plans of the 
squares A and B. As neither of these cubes appears to touch the plane 
of the picture M M, it will be necessary to imagine the sides I </, and h A, 
to be continued until they do so ; now draw down perpendicular lines from 
where the continuations of these sides intersect the base line, and set off 
on them from the line M'M', the height of the cube, as 1 — 2 which 
is the same as the width, and complete the square shown by the dotted 
lines : from all four angles of this square draw lines to the centre of view 
— this will give the representation of four lines at right angles with the 
picture carried on as far as it would be possible to see them ; then it only 
remains to cut off the required perspective widths of the cubes by \\\q per- 
pendicular lines from the intersection of the visual rays with the plane of 



PLATE XCIX. 



890 




PERSPECTIVE DRAWING. 391 

the picture : tlie completion of this problem will be very easy, if the 
drawing of the squares is well understood. 

In such simple objects as these it will not be necessary to draw a plan ; 
when one side is parallel to the picture, and dimensions are known. In 
fig. 2, the same objects as those in fig. 1 are drawn without a plan thus : — 

Draw the ground line M IM, then the vertical line S' Y, and the horizon- 
tal line D D, at the height of the eye ; making D D the same distance on 
each side of Y, that the eye is from the transparent plane ; for drawing 
the squares mark off from S' to h\ on the ground line, the distance that 
the square is on one side of the observer ; let h' a' be the length of one 
side of the square ; from V and a' draw lines to Y, which represent the 
sides of the square carried on indefinitely ; to cut off the required per- 
spective width of the side V e' of the square, lay off' the width, a' h\ from 
b' to j9, then draw from jj> to D on the left, and the point e' where the line 
Dj»j> intersects h' Y, will give the apparent ^\ddth required ; then draw/" e 
parallel to a' V^ and the square is complete : this may be proved in the 
same way as in fig. 1. The further square may be obtained in a similar 
manner, setting off the distance between the squares fromj9 to ^, and the 
width of the square beyond that, and drawing lines to D as before : 
some of the lines in this plate are not continued to the ground line, 
in order to avoid confusion. Proceed with the cnbes by the same rule. 
Let 1, 2, 3, 4, be the size of one side of the cube if continued until touch- 
ing the picture ; from these points draw rays to Y : from 2> iQ> t set off the 
distance the cube is from the picture, and from t to t^ the width of the 
cube ; draw from these points to D on the right, and their intersection of 
the line 3 Y in??z, 6>, will give the perspective width and position of that 
side of the cube : draw lines perpendicular to the ground line from m and 
(9, and lines parallel to 4 — 2 from the angles of the cube, Z', (j\ m ; then 
draw the side n Ji, and the cube is complete. Tlie operation of drawing 
the other cube is similar, and easy to be understood.y 

From the drawing of a square in parallel perspective, we deduce rules 
for the construction of a scale in perspective. Let D M M D, (fig 6,) be 
the plane of the picture, the same letters of reference being used as in 
preceding figures. From S' lay off the distance o S' equal to some unit 
of measure, as may be most convenient ; from o draw the diagonal to D 
the point of distance ; now draw 1 1' parallel to the ground line M M, 
again draw from V the diagonal V D, and lay off the parallel 2 2', pro- 
ceed in the same way with the diagonal 2' D and the parallel 3 3', and 
extend the construction as far as maybe necessary. It is evident o^' 11\ 
1^12 2', 2' 2 3 3' are the perspective projectors of equal squares, and 



392 



PEKSPECTIVE DEAWING. 



therefore o S', 1 V, 2 2' 3 3', etc., and S' 1, 1 2, 2 3, etc., are equal to each 
other, and that if o S' is set off to represent any unit of measure, as one 
foot, one yard, or ten feet, &c., each of these lines represents the same dis- 




Fig. 6. 

tance, the one being measures parallel to the base line, the others perpen- 
dicular to it. In making a perspective drawing a scale thus drawn will be 
found very convenient ; but as in the centre of the picture it might inter- 
fere with the construction lines of the object to be put in perspective, it is 
better that the scale be transferred to the side of the picture <^ M ^, the di- 
agonals to be laid off to a point to the right of D equal to the point of 
distance. 

The scales thus j)rojected are for lines in the base or ground plane ; for 
lines perpendicular to this plane the following construction is to be adopt- 
ed ; upon any point of the base line removed from S^, as a for instance, 
erect a perpendicular, ad i on this line, lay off as many of the units o S' 
as may be necessary ; in this example three have been laid off, that is, a d 
=3 S^ Erom a and d draw lines to the centre of view, and extend the 
parallels 1 1', 2 2', 3 3 ' ; at the intersection of these lines with a Y erect 
perpendiculars. The portions comprehended between the lines a Y and d 
Y will be the perspective representations of the line a d, in planes at dis- 
tances of 1, 2, 3, 6> W from the base line, and as Z>, c, d are laid off at inter- 
vals equal to o^\ by drawing the lines c Y and h Y six equal squares are 
constructed, of which the sides correspond to the unit of measure, o S'. 

To determine the Persjpective Position ofanyjpoint in the Ground Plane. 
Thus (fig. 7), to determine the position of the point j?, which in plane 
would be six feet distant from the plane of the picture, M M, and ten feet 
from the central phxne, to tlie left. 

Lay off from S', to the left, the distance a S', equal to six feet on the 



PERSPECTIVE DRAWING. 



393 



scale adopted; draw the diagonal to the point of distance D, on the right; 
at its intersection a' with tlie vertical line Y S', draw a parallel to M M 
the base line; lay off from S', S' h equal to ten feet, draw h Y ; the inter- 
section of this linej;, with the parallel previously drawn, will be the posi- 
tion of the point required. 



D 








7^ 




D 






I 


P^ 


^^^^ 


S 


-""^ 






M 


; /-^"^ 


I 


"^^/^--^'^^ 


ill 






/ 


/ 


-^^ 






J 


^'^ 





M 



Fig. T. 



M 



By a similar construction the position of any j)oint in the ground plan 
may be determined. It is not necessary that the distances should be ex- 
j^ressed numerically ; they may be shown on the plan and thence be trans- 
ferred to the base line, and thrown into perspective by the diagonals and 
parallels. As the intersections of the various lines of the outlines of ob- 
jects are points, by projecting perspectively these points, and afterwards 
connecting by lines, the perspective of any plane surface, on the ground 
plane, may be shown. 

If the point ^ were not in the ground plane, but in a position directly 
above that already assumed, that is, the distances from the plane of the j)ic- 
ture, and the central plane being the same, but its distances above the 
ground plane were, say, five feet ; then at h erect a perpendicular, and lay 
off h y equal to five feet, connect h' Y, at j) erect another perpendicular, 
and its intersection ]/ w^th the line V Y will be the position of the point 
required. 

Or the plane of the point f' might be assumed as the position of the 
ground plane, M' M' becoming the base line, and laying off from S''', S''' a!' 
andS'''^' — equal respectively to six and ten feet; drawing the diagonal 
o!'J) and VN and the parallel as before, the point j^' will be determined. 

To draw an Octagon in Parallel Perspective. — Let A (fig. 8) represent 
the plan of an octagon. Draw M M, S' Y, and D D, as before ; from the 
points M, (2, J, c, draw rays to Y. Set off on M M from c to the right the 
distances ce^ cd^ cf^ from wdiich draw diagonals to D on the left, and 
at their intersection w^ith the ray c Y, draw parallels e' g\ cT h\ h' l\ to 
the base line ; these points will correspond to the angles on the plan. JSTow 



394 



PEESPECTIVE DEAWIKG. 



connect tlie angles on the perspective view, in the proper succession, and 
the perspective projection is complete. 

It will be observed, that in this construction the plan has been placed 
forward of the plane of the picture, contrary to the position it should oc- 
cupy, which should be the same relative position back of this plane ; but 
it will be found much simpler in construction than if it were placed as in 
PL XCIX. and the points were all projected to the base line ; it is, of course, 
equally correct in its perspective projection. 



h- I 




l^/ 



M 



S ly 





Fig. 8. 



To draw a Circle in Parallel Ferspective.—Let C, (fig. 8) represent the 
plan of a circle, round which let the square aec7n be described, two of 
its sides being parallel to the base line M M ; draw diagonals across the 
square, and where these intersect the circumference of the circle draw the 
lines I) Jc and dg parallel to the base line, and the lines o n and^ </ at right 
angles thereto. Draw also the lines /Z and ch at right angles to each 
other through the centre of the circle, project the points a, o, l,p, m., to 
the base and draw rays to Y ; set off from a' to the left the distances a' a, 
ah, a'c, a'd, a'e, and draw diagonals to the point of distance D on the 
right ; at their intersection with the line oJ V draw horizontal lines, or 
parallels to the base, and there will be projected in perspective the square 
aec m, witli all the lines of parallels and perpendiculars ; connect the 
intersections corresponding to the points c, n, /, (/, //, A*, I, 



?•, and we 



PERSPECTIVE DRAWING. 



305 



have the perspective projection of the required circle, wliich will be an 
ellipse. 

To erect upon the octagonal base A an octagonal j^illar or tower. This 
construction resolves itself into simply constructing another octagon on an 
upper plane, and connecting the visible angles by perpendiculars, or per- 
pendiculars may be erected at the points M, a^ ^, <?, and the heights of the 
tower laid off npon them, and from these extremities rays drawn to the 
centre of view ; the intersection of these rays by perpendiculars from the 
angles of the octagon beneath will determine the j)rojection of the upper 
surface of the pillar ; represent in full lines all visible outlines, and the 
projection is complete. 

In the same manner a pillar may be erected on the circular base. If 
the pillars be inclined, the first method of projecting the upper outline on 
a plane assumed at the height of the pillar, must be adopted. 

To draw a Pyramid in Parallel Persjpective. — Let A (fig. 9) be the 
plan of a pyramid, the diagonal lines represent the angles, and their in- 
tersection the vertex; project the plan as in previous examples of squares. 
Draw diagonal lines from M to 5, and a to <?, their intersection gives the 
perspective centre of the square ; upon this point raise a perpendicular 
line which is the axis of the pyramid ; draw a perpendicular line e /*, in 




Fig. 9. 



the centre of the line M a^ upon which set up the height of the pyramid 
ef ; from/* draw a line to Y, and its intersection of the axis of the j^yramid 
at d will give the persj^ective height ; complete the figure by drawing 
lines from d^ the apex, to M, <2, J, the three visible angles. The other 
two pyramids are drawn in a similar manner, by setting their distances 
from the plane of the picture off from (X, on the ground line to the right, 
and drawing diagonals to the point of distance on the left. 



396 PEESPECTIVE DRAWING. 

To draw a Cone in Parallel Perspective. — Let B (fig. 9) represent the 
plan of a cone, apply the same lines of construction as to C (fig. 8) ; and 
draw the perspective view of a circle, upon the perspective centre of 
which draw a perpendicular line, a, h ; on the centre of the line d e^ raise 
a perpendicular, upon which set up the height of the cone, from the 
ground line to c / from c draw a ray to Y, and the point where this line 
intersects the axis of the cone a 5, in Z», will give the perspective height 
of the axis ; from h draw lines to/* and ^, and the figure is complete. 

To draw the prism, C, which consists of two triangular ends and three 
rectangular sides, place the length of the side a M upon the ground line, 
and draw lines to Y ; mark oft' the width of one end from a to the left upon 
the ground line, and draw to the point of distance on the right, which 
gives the perspective width, ad; find the perspective centre /" of the 
side in the same way, and from f and d draw horizontal lines until they 
intersect the line from M Y ; upon f and g draw perpendiculars ; set up 
the height a Z>, of the end of the prism, and from h draw a line to Y, and 
the point where it intersects f c^ in c, will give the perspective height of 
the end of the figure ; from c draw c e^ parallel to a M, from c draw c d, 
and c a J and the visible end is complete ; the other end is dotted in to show 
the process only. 

E is the representation of a cylinder, with one end towards the spec- 
tator ; its projection will be easily understood by examination. 

To draw a Square and Cttbe in Angular Persjpective. Plate C. Let 
A (fig. 1) be the plan of the square, and B the plan of the cube, M M the 
base or ground line, and S the station point. Draw M' M', and D D' par- 
allel to M M, flie one being the ground line and the other the horizon of 
the plane of the picture ; project the point cZ on M M to dJ^ on M' M'. It 
has been shown in parallel perspective that the vanishing points of diago- 
nals of squares lie in the points of distance ; if through the station point 
S, in any of the preceding figures, lines be drawn parallel to the diagonals, 
tliey will intersect the base lines at distances from the central plane equal 
to the points of distance. Li like manner to find the vanishing points of 
lines in the ground planes, or in planes parallel to the ground plane, in- 
clined to the plane of the picture, through the station point S draw lines par- 
allel to tlie inclined lines, and project their intersection with the base line 
to the horizon of the picture ; thus, in the j)rcsent example draw S M, S JVI 
parallel to ad^ e A, and to dc^ kg j project their intersections M, M, with 
the base lino to ]^, I)', the horizon of the picture, and D, D\ will be the 
vanishing points of all lines parallel to a d and d c. Draw d' D and d' D', 
the perspective projection of da will lie in the former of these lines and 



PLATE C. 



dim; 




PERSPECTIVE DRAWING. 397 

d c in the latter. To determine the perspective position of the points a 
and c, or the length of these lines, draw the rays a S and c S, project their 
intersection with tlie base M M, upon the lines d' D and d' D', and their in- 
tersections a\ d will be the perspective projection of the points a and c. 
To complete the projection of the square, draw the lines a! D' and c D, 
their intersection will be the perspective projection of the point Z>, and the 
square is complete. To prove tlie construction, draw tl^e ray h S and pro- 
ject its intersection with the base M M, and if the construction be correct 
it will fall upon the point V . 

As the cube is placed at some distance from the plane of the picture, 
it will be necessary to continue either eh ov g A, or both, till they intersect 
the base line M M at n and 7?^/ drop pei'pendiculars or project these points 
upon M M' at n' and m' ; on these perpendiculars set up the height of the 
cube m' o and oi' s, draw the lines m' D', o' D^ and n' D, sD ; connect the 
intersections h' and h" ; draw the rays S e and S g, and project their inter- 
sections with M M, to g^ e' / draw the lines e" J)' and ^" D ; if the construction 
be correct, the projection of the intersection of the ray Sywith the base 
will fall upony^, and of the ray S A will fall upon h!' and li' . 

To Solve the Same Problem hj a Different Constricction. — Let A and 
B, (fig. 1,) be as before the plans of the square and of the cube ; to ])ro- 
ject them perspectively on the plane of the picture MD D'M, (fig. 2). 

From the point M and M, (fig. 1,) set off distances equal to M S, M S, 
to jj> and^;>' / project these points upon D D' fig. 2, the point ^;>, fig 2, will be 
that from which any number of parts may be laid ofiT on lines vanishing in 
D'; the point J? will be the corresponding point for lines vanishing in D. 
These points may be called the points of division. In parallel perspec- 
tive the points of distance were the points of division, the one for the 
other. To illustrate their application in the present example, project the 
point c7, (fig. 1,) to d' (fig. 2,) draw d' D and d' D', from d' on either lay 
off a distance d' ?', d' h equal to the side of the square a d. Kow since j? 
is the division point of lines vanishing in D from i^ draw the line ip, and 
its intersection w^itli d' D cuts off a line d' a' equal perspectively to the 
line d' i or a d measured on the base line. Again since j^' is the division 
point of lines vanishing in D^, the line Icj^' cuts off on d T)\ a line d' c' equal 
perspectively to the line d' h^ov ad measured on the base : having a' d' c, 
the square is completed by drawing the lines c' h' towards D, and a' 1) 
towards D'. 

To construct the cube, project the point m, (fig. 1.) to m', (fig. 2) ; lay 
off on the perpendicular forming the projection, the height ni' o' of the 
cube ; draw the lines vi' D' and o' J)' . Lay off the distance m' r equal to 



398 PEESPECTIVE DE AWING. 

m h, (fig. I5) and draw the line rp\ its intersection with m' T>' will cut off 
7n' h\ eqnal to m A, (fig. 1,) and establish the angle h of the cube. From 
T lay off r ^, equal to Jig^ (fig. 1,) draw s ;p\ and its intersection with m' D^ 
establishes the angle g'. From h' draw a line vanishing in D. Through 
h' extend a line j? N to ?5, from z^ lay off to the left t a, equal to the side of 
the cube h e ; draw a^^ and its intersection with the line N D, establishes 
a third point e of the cube. Upon these points N g' e' erect perpendicu- 
lars ; those upon li' and g' will, by their intersection with 0' D, determine 
7/ g" . Draw ¥ D', its intersection with the perpendicular at c determines 
e". Draw g" D' and ^' D to their intersection, and the cube is complete. 

To Draw the Perspective Projection of an Octagonal Pillar in Angu- 
lar Perspective. — Plate CI. Let A, (fig. 1,) be the plan of the pillar. En- 
close it by a square. Let M M be the base line, and S the station point ; de- 
termine the position of the vanishing points for the sides of the square as in 
Plate C, and project the square upon the plane of the picture M D D^ M' 
by either of the methods already explained. These lines of construction are 
omitted, as on the necessarily small diagrams they w^ould confuse the stu- 
dent ; but in drawing these examples to the scale recommended, they might 
be retained. From the angles of the octagon visible to the spectator 
draw rays to the station point S project their intersection with the base 
line MM, to the perspective square, (fig. 2,) which will thus determine on 
the sides of the square the positions of the points co\ V^ c\ d' ^ e\ correspond- 
ing to the visible angles of the octagon ; connect these points by lines. To 
construct the pillar upon this base, upon the perpendicular let fall from 
the corner/" of the square upon M M^ at/ set off the height of the pillar ; 
from this pointy draw lines to the vanishing points D, D^, and construct 
three sides of an upper square similar to the lower one. The lines of this 
square will determine the length of the sides of the tower, which are the 
perpendiculars let fall upon d V c d' e' . 

To Construct a Circular Pillar in Angular Persjpective. — Plate CI. 
Let B, (fig. 1,) be the plan of the base ; enclose it with a square whose sides 
are parallel respectively to S M and S M ; project this square upon the plane 
of the picture, (fig. 2,) divide the plan into four equal squares by lines par- 
allel to the sides ; draw rays through the points h and % and project their 
intersection witli M M upon the perspective square. From the points N and 
i' thus formed, draw lines to vanishing points D' and D, and the perspec- 
tive square is divided similarly to the original, and there are four points of 
the circle established : through these draw the perspective of the circle. 
P)y the division of the base into smaller squares more points of the curve 
might be determined, but for the present purpose they are unnecessary. 



PLATE CI. 




Fi- 1. 



Fifr. 2. 




^Jf 




PERSPECTIVE DKAWIXG. 399 

To determine the outline of the i)ilhir, draw from S rays tangent to the 
sides of the phm at k and /, the perpendicuhirs let fall from their intersec- 
tion with M M will he the outline of the cylinder. To cut them off to the 
proper height, and to determine the top of the cylinder, upon the perpen- 
dicular let fall upon i^ set off the height of the cylinder V l\ and upon this 
plane project the square as before, and draw in through the points thus 
determined the outline of the curve. As a still further elucidation of the 
principle of projection, an enlarged cap is represented on the pillar, of 
which the circumscribing circle (fig. 1,) is the plan. In this by extending 
the central lines of the square, both in plan and perspective, we are en- 
abled to project readily eight points in the larger circle through which 
the curve may be drawn. 

To Draw an Octagonal Pyramid in Angular Persjpective. — Plate CI. 
Let/", (fig. 1,) be the base of the pyramid ; project upon the plane of the 
picture, (fig. 3,) the visible angles of the base, as in the case of the pillar. 
Through the centre of the plan draw a line parallel to one of the sides and 
intersecting M M at on / from this point let fall a perpendicular to m' on 
MM', (fig. 3,) ; on this perpendicular set ofi:' the height of the pyramid 7n 
from rn' and draw lines to D'. From the centre of the plan draw a ray 
to S, and project its intersection with M M, upon the line o T)\ its intersec- 
tion o' with this line will be the apex of the pyramid : from this point 
draw lines to the angles of the base already projected, and the pyramid is 
complete. 

To Draw a Cone in Angular Persjyective. — Plate CI. Let the inner 
circle B, (fig. 1,) be the base of the cone j^roject its visible outline to fig. 
3, as in case of the cylinder. To determine its height extend one of the 
diameters of the plan to the base line at^; from this point let fall a per- 
pen-dicular toj;' on MM', and set oflf ujDon itj?'^/, the height of the cone; 
from^;>' and g draw lines to the vanishing point D'. From the centre of 
the i^lan, (fig. 1,) draw rays to S, and project its intersection with MM, 
upon r' on the line g D', and r' will be the apex of the cone : connect the 
apex with the extremities of the perspective of the baseband the projec- 
tion of the cone is complete. 

To Drav: the Elevation of a Building in Angular Persjoective. — Plate 
CII. For example, take the school-house, PL LXXIY. of architecture. 
Plot so much of the plan of the building at it as may be seen from the po- 
sition of the spectator at S. Draw a base line, and through the station 
point draw parallels to the sides of the building cutting the base as at M 
M: draw MM' for a base, and DD' for the horizontal line of the j)icture. 
Project M and M to D and D', for the vanishing points, the one of the 



400 PEESPECTIVE DRAWING. 

lines parallel to a c, the other to aJ) j extend a c^ ah ', project d^ e^ to 
d\ e\ and on d' d set off the height of the eaves d^ o^ and of the ridge d 
n / from d\ o and oi draw lines to T>\ and from e to D, draw rays from c 
and h to S^, and project their intersection with the base to the vanishing 
lines just drawn. To find the perspective of the ridge draw a ray from 
the centre of a 5, and project its intersection with the base to r on the line 
01 J}\ the point is the apex of the gable, the line r D will be the perspective 
of the ridge ; to determine its length erect a perpendicular at the intersec- 
tion of t T>' and s D, draw the sloping lines of the roof, and the outline of 
the building is complete. The filling in of the details will be readily under- 
stood ; it will only be necessary to keep in mind, that all lines parallel to 
a 1) must meet in D^, those io acvciY)'. all measures laid ofiT on any lines 
of the plan must be connected with the point of sight S, and their inter- 
sections with the base projected. All vertical heights must be laid off on 
the line d' d, and referred to the proper position by lines to D or J)\ as the 
case may be. 

As an example of the other method of constructing this same problem, 
let the scholar lay ofi" to the double of the present scale the plane of the 
picture M D D^ M^, and the division points^' and^, and without drawing 
plan or elevation take the dimensions from Plate LXXIY. 

To Draw an Arched .Bridge in Angidar Perspective. — PL CIII. Let 
A and B, (fig. 1,) be the plans of the piers; on the line a A, one of the sides 
of the bridge, lay down the curve of the arch as it would appear in eleva- 
tion, in this example an ellipse. Divide the width of the arch as at h. c. 
d, e.f. g. A., carry up lines perpendicular to h h until they intersect the 
curve of the arch, and through these points, draw lines parallel to 5 A as Tc. 
I. m. / let (9 ^ be the height of the parapet of the bridge above the spring of 
the arch. Through th6 station point draw lines parallel to the side a h 
and end a a of the bridge, till they intersect the assumed base line M M : 
project these intersections to the horizon line of the picture for the vanish- 
ing points D, D^ of perspective lines parallel ioahimdiaa. Let fall a 
perpendicular from a to a^^ and on this perpendicular set ofi" from a' tlie 
heights s \ si, s m, and st ; from a' and r' draw lines to D and D', and from 
the points w/, l\ h' to J)'. Draw rays from the j^oints a. h. c. d. e.f. g. h. 
to tlie station point S, and project their intersection with tlie base lines to 
the perspective line a' D' as in previous examples : the intersection of the 
lines Ic' D', V D', on' D' by the perpendiculars thus projected, will establish 
the points of the curve of the arch on the side nearest the spectator. To 
determine the j^osition of the opposite side of the arch, from a", the per- 
spective width of the bridge, draw a" D', and from h' draw lines to D ; 



PLATE Cir. 




4ro 



PLATE cm. 




PERSPECTIVE DRAWING. 401 

the line h' j^' will be the perspective width of the pier ; draw y D ; and from 
h\ ¥ D ; from g" the intersection of the curve of the arch by the perpen- 
dicular to g\ draw g" D, the intersection with h" D' will be one point in 
the curve of the arch on the opposite side of the bridge : in the same way, 
from any point in the nearer arc draw lines, to D, and the intersection 
wdth lines in the same planes on the opposite side of the bridge, Avill fur- 
nish points for the further arch : all below the first only >Yill be visible to 
the spectator. 

To Draw m Parallel Perspective the Interior of a Room. — PL CIII. 
We pro230se to construct this by scale without laying down the plan. 
Draw the horizon line D Y D', and the base M M', making D and D' the 
point of distance. Let the room be 20 feet wide, 14 feet high, and 12 feet 
deep ; on the base M M^, lay off the rectangle of the section in our figure 
on a scale of 8 feet to the inch, 20 feet x 14 feet. From the four corners 
draw lines to the centre of view Y ; from S' lay off to the right or left on 
M M^ 12 feet, and through this point draw lines to D' or D as the case may 
be : through the point of intersection a' of this line with S' Y draw a line 
parallel to M M' ; at the intersections of this line with M Y and M' Y erect 
a perpendicular, cutting the vanishing lines of the upper angle of the 
room at d and e ; connect d e and the perspective of the room is complete. 
To draw the aperture for a door or window on the side, measure off from 
S' the distance of the near side from the plane of the picture, and in addi- 
tion thereto the width of the aperture ; from these two points draw lines to 
the proper point of distance, and at their intersection w4th S' Y, draw j^ar- 
allels to M M^, cutting the lower angles of the room, and erect perpendic- 
ulars, the height of which w^ill be determined b}^ a line drawn from/*, the 
height of the window above the floor measured on M D. Should the ^vdn- 
dow be recessed, the farther jamb will be visible ; extend the farther par- 
allel to M M', and cut it by a line gY. M ^ being the depth of the recess, 
the rest of the construction may be easily understood by inspection of the 
figure. At the extremity of the apartment a door is represented half open, 
hence as the plane of the door is at right angles to the plane of the picture, 
the top and bottom lines will meet in the point of view ; if the door were 
open at an angle of 45°, these lines would meet in the points of distance ; 
if at any other angle, the vanishing points would have to be determined 
by constructing a plan, drawing a line parallel to the side of the door 
through the station point, and projecting it upon the horizon line. Tlie 
chair in the middle of the room is placed diagonally, and the table parallel 
to the plane of the picture ; their projection is simple. 

To Draw in Persjyective a Flight of Stairs. — PL CIY. Lay off the 
26 



402 PEESPECTIYE DRAWING. 

base line, horizon, centre of view, and point of distance of the picture, 
construct the solid ah c d^ efg h^ containing the stairs, and in the required 
position in the plane of the picture, divide the rise a c into equal parts ac- 
cording to the number of stairs, four for instance ; divide perspectively the 
line a h into the same number of parts ; at the points of division of this 
latter erect perpendiculars, and through the former draw lines to the cen- 
tre of view ; one will form the rise and the other the tread of the steps. 
From the top of the first step to the top of the upper continue a line a d, 
till it meets the perpendicular S^ Y prolonged in v / this line will be the 
inclination or pitch of the stair ; if through the top of the step at the other 
extremity a similar line be drawn, it will meet the central perpendicular 
at the same point v^ and will define the length of the lines of nosing of 
the steps, and the other lines may be completed. As the pitch lines of both 
sides of the stairs meet the central vertical in the same point, in like man- 
ner V will be the vanishing point of all lines having a similar inclination 
to the plane of the picture. The projection of the other fiight of stairs 
will be easily understood from the lines of construction perpendicular to 
the base line or parallel thereto, lying in planes. 

To Find the Beflection of Objects in the Water. — PL CIY. Let B be 
a cube suspended above the water ; we find the refiection of the point a^ 
but letting fall a perpendicular from it, and setting off the distance a' w 
below the plane of the water equal to the line a w above this line ; the line 
wf will also be equal to the line wf^ find in the same way the points V 
and e\ through these points construct perspectively a cube in this lower 
plane, and we have the reflection of the cube above. 

To find the refiection of the square pillar D removed from the shore : 
suppose the plane of the water extended beneath the pillar, and proceed 
as in the previous example. 

It will be observed that those lines of an object which meet in the cen- 
tre of view Y, in the original ; their corresponding reflected lines will con- 
verge to the same point. If the originals converge to the points of distance, 
the reflected ones will do the same. To find the refiection of any inclined 
line, find the reflection of the rectangle of which it is the diagonal, if the 
plane of the rectangle is perpendicular to the plane of the picture ; if the 
line is inclined in both directions enclose it in a parallclopidcd and project 
tlie reflection of the solid. 

To find the Perspective Projection of Shadows. — Plate CY. Let the 
construction points and lines of the picture be plotted. Let A be the per- 
Rpcctivc projection of a cube placed against another block, of Avhicli the 
face is parallel to the phme of tlie picture : to find the shadow upon tlie 



PLxVTE CIV 




PLATE CV. 



402 




PERSPECTIVE DRAWING. 403 

block aud upon the groimd plane, supposing the light to come into the 
picture from the upper left-hand corner and at an angle of 45°. Since the 
angle of light is the diagonal of a cube, construct another cube similar to 
A, and adjacent to the face d c g ; draw the diagonal 11'^ it will be the 
direction of the rajs of light, and Ic will be the shadow of I ; connectyX* 
and cJc^fh must be the shadow of the line hf^ and ck oi h c ; the one 
upon the horizontal plane and the other in a vertical one : the former will 
have its direction, being a diagonal, toward the point of distance D', the 
other being a diagonal in a j^lane, parallel to tliat of the picture, will be 
always projected upon this 2:)lane in a j)arallel direction. 

Let B be a cube similar to A; to find its projection upon a horizontal 
plane, the shadow of the jDoint V may be determined as in the preceding 
example, but the shadow of the point c\ instead of falling upon a plane 
parallel to the picture, falls upon a horizontal one ; its position must be de- 
termined as we did before by h. Construct the cube and draw the diag- 
onal c' I ; in the same way determine the point m' the shadow of cT ; con- 
nect ch' Ira n^ and we have the shadow of the cube in perspective on a 
horizontal plane. 

On examination of these projected shadows, it will be found that as the 
rays of light fall in a parallel direction to the diagonal of the cube, the 
vanishing point of these rays will be in one j^oint T^ on the line D^ M' 
prolonged, at a distance below D' equal YD'; and since the shadows of 
vertical lines upon a horizontal plane are always directed towards the point 
of sight, the extent of the shadow of a vertical line may be determined 
by the intersection of the shadow of the ground point of the line by the 
line of light, from the other extremity. Thus, tlie point 7t, cube A, is the 
intersection oif D' by 5 Y^ ; the points Z;', Z, m are the intersections of c D', 
o D', n D' by V Y^, c' Y'by cZ' Y.' Similarly on planes parallel to that of the 
picture, Z', cube A is intersection of the diagonal c Z,', by the ray of light t Y'. 

Applying this rule to the frame C, from r, s^ i)^ draw lines to D' from ;■', 
s\p\ draw rays to Y'; their intersections define the ontline of the shadow of 
the post. To draw the shadow of the projection, the shadow uj)on the post 
from t will follow the direction of the diagonal c Jc. Project u and v upon 
the ground plane at u' and v' j from t' u' v' andj? draw lines to D'; from t. 
v., ^', IV and x draw rays to Y', and the intersection of these lines with their 
corresponding lines from their bases will give the outline required ; as v and to 
are on the same perj^endicular, their rays will intersect the same line v' Y'. 

With reference to the intensity of '* shade and shadow " and the neces- 
sary manipulation to produce the required efiect, tlic reader is referred to 
the article on this subject. 



404: PERSPECTIVE DRAWING. 

In treating of Perspective it lias been considered not in an artistic 
point, as enabling a person to draw from nature, but rather as a useful art to 
assist the architect or engineer to complete his designs, by exhibiting them 
in a view such as they would have to the eye of a spectator when con- 
structed. In our examples, owing to size of the page, we have been limit- 
ed in the scale of the figures, and in the distance of the point of view, or 
distance of the eye from the plane of the picture, and as it was unimportant 
to the mathematical demonstration, few of the figures extend above the 
line of the horizon. In these particular points it is unnecessary that the 
examples should be copied. The most agreeable perspective representa- 
tions are generally considered to be produced by fixing the angle of vision 
M S M^, at from 45 to 50°, and the distance of the horizon above the 
ground line at about one-third the height of the picture. 

Linear perspective is more adapted to the representation of edifices, 
bridges, interiors, &c., than to that of machinery ; it belongs, therefore, 
rather to the architect than to the engineer or the mechanic ; for the pur- 
poses of the latter we would recommend Isometrical Perspective; uniting 
accuracy of measures with graphic perspective representation. 



I60METRICAL DKAWINO. 



405 



ISOMETKICAL DEAWmG. 




Peofessor Faeish, of Cambridge, has given the term Isometrical Per- 
spective to a particular projection which represents a cube, as in fig. 1. 
The words imply that the measure of the representations of the lines 
forming the sides of each face are equal. 

The principle of isometric representation con- 
sists in selecting for the plane of the projection, 
one equally i'nclined to three principal axes, at 
right angles to each other, so that all straight lines 
coincident with or parallel to these axes, are drawn 
in projection to the same scale. The axes are 
called isometric axes, and all lines parallel to them 
are called isometric lines. The planes containing 
the isometric axes are isometric planes ; the point 
in the object projected, assumed as the origin of the axes, is called the reg- 
ulating point. 

To draw the isometrical projection of a cube, (fig. 2,) draw the hori- 
zontal line A B indefinitely ; at the j^oint D erect the perpendicular D C, 
equal to one side of the cube required ; through D draw the line D h and 
T>f to the right and left, making fDB and hJ) A each equal an angle of 
30°. Consequently the angles P D/* and F D J are each equal to 60°. 
Make D h and Df each equal to the side of the cube, and at h and/* erect 
perpendiculars, making h a and f e each equal to the side of the cube ; 
connect P a and P e and draw e g parallel to a P, and a g parallel to P ^, 
and we obtain the projection of the cube. 

If from the point P, with a radius P D, a circle be described, and com- 
mencing at the point D radii be laid ofiF around the circumference, forming 
a regular inscribed hexagon, and the points T) a e 'bQ connected with the 



406 



ISOMETRIC AL DRAWING. 



centre of the circle F, we have an isometrical representation of a cube. 
The point D is called the regulating point. 

K a cube be projected according to the principles of isometrical per- 
spective, in a similar manner as we have constructed one according to the 
rules of linear perspective, the length of the isometrical lines would be to 
the original lines as .8164 to 1, but since the value of isometrical perspec- 
tive as a practical art lies in the applicability of common and known 
scales to the isometric lines, in our constructions we have not thought it 
necessary to exemplify the principles of the projection, but have drawn our 
figures without any reference to what would be the comparative size of the 
original and of the projection, transferring measures directly from plans and 
elevations in orthographic projections, to those in isometry. It will^be ob- 
served that the isometric scale adopted applies only to isometric lines, as 
FD,¥a and F 6 or lines parallel thereto ; the diagonals which are abso- 
lutely equal to each other, and longer than the sides of the cube, are the 
one less, the other greater ; the minor axis being unity, the isometrical lines 
and the major axis are to each other as, 1. ^2. |/3. 

Understanding the isometrical projection of a cube, any surface or 
solid may be similarly constructed, since it is easy to suppose a cube suffi- 
ciently large to contain within it the whole of the model intended to be 
represented, and as hereafter will be farther illustrated, the position of any 
point on or within the cube, the direction of any line or the inclination of 
any plane to which it may be cut, can be easily ascertained and repre- 
sented. 

In figs. 1 and 2 one face of the 
cube appears horizontal, and the 
other two faces appear vertical. If 
now the figures be inverted, that 
which before appeared to bo the top 
of the object, will now appear to 
be its under side. 



1 




The angle of the cube formed 
by the three radii meeting in the 
centre of the hexagon, may be 
made to appear either an internal or 
external angle ; in the one case the 
faces representing the interior, and 
in the otlicr the exterior of a cube. 
Figs. 3, 4, 5, illustrate tlie application of isometrical drawing to simple 



ISOMETRICAL DRAWING. 



407 



combinations of the cube and parallelopipedon. Tlie mode of construction 





Fiss. 3. 4, 5. 



of these figm'es will be easily understood by inspection, as they contain no 
lines except isometrical ones. 

To draio Angles to the Boundary Lines of an Isometrical Cvhe, 




30 20 iO 



Fig. 6, 7. 



Draw a sqnare C (fig. 6,) wliose sides are equal to those of the isome- 
trical cube A, and from any of its angles describe a quadrant, which di- 
vide into 90°, and draw radii through the divisions meeting the sides of 
the square. These will then form a scale to be applied to the faces of the 
cube ; thus on D E, or any other, by making the same divisions along their 
respective edges. 

As the figure has twelve isometrical sides, and the scale of tangents 



408 



ISOMETEICAL DRAWING. 



may be applied two ways to eacli, it can be applied therefore twenty-four 
ways in all. "We thus have a simple means of drawing, on the isometrical 
faces of the cube, lines, forming any angles with their boundaries. 

Figs. 1, 2, 3, 4, 5, 6, PL CYI. show the section of a cube by single 
planes, at various inclinations to the faces of the cubes. Figs. 7, 8 are the 
same cube, but turned round, with pieces cut out of it. Fig. 9 is a cube 
cut by two planes forming the projection of a roof. Fig. 10 is a cube with 
all of the angles cut o& by planes, so as to leave each face an octagon. 
Fig. 11, represents the angles cut off by planes perpendicular to the base 
of the cube, forming thereby a regular octagonal cylinder. By drawing 
lines from each of the angles of an octagonal base to the centre point of 
the upper face of the cube, we have the isometrical representation of an 
octagonal prism. 

As the lines of construction have all been retained in these figures, 
they will be easily understood and copied, and are sufficient illustrations 
of the method of representing- any solid by enclosing it in a cube. 

We have now to consider the application of this species of projection 
to curved lines. 



D 




Figs. 8, 9. 



Let A B (fig. 8,) be the side of a cube with a circle inscribed : and 
suppose all the faces of the cube to have similarly inscribed circles. Draw 
the diagonals A B, CD, and at their intersection with the circumference 
lines parallel to A C, B D. Now draw the isometrical projection of the 
cube, (fig. 0,) and lay out on the several faces tlic diagonals and the par- 
allels ; tlie projection of the circle will be an ellipse, of which the diago- 
nals being the axes, their extremities are defined by their intersections/ G, 



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409 



PLATE CVII. 




PLATE CYIII. 




ISOMETKICAL DKAWING. 



409 



e5, 



a 2, hi, dS, c4:, by the parallels; having thus the major and minor 
axis, construct the ellipse by the trammel, or since the curve is tangent at 
the centre of the sides, Ave have eight points in the curve ; it may be put 
in by sweeps or by the hand. 




Fis. 10. 



To Divide the CivcumfeTence of a Circle. — First method. On the 
centre of the line A B erect a perpendicular C D, making it equal to C A 
or C B ; then from D, with any radius, describe an arc and divide it in 
the ratio required, and draw the divisions radii from D meeting A B ; 
then from the isometric centre of the circle draw radii from the divisions 
on A B, cutting the circumference in the points required. 

Second method. On the major axis of the ellipse describe a semi-cir- 
cle, and divide it in the manner required. Through the points of division 
draw lines perpendicular to A E, which will divide the circumference of 
the ellipse in the same ratio. On the right hand of the figure both 
methods are show^n in combination, and the intersection of the lines give 
the points in the ellipse. 

Plate CYII., fig. 1, is an isometrical projection of a common bevel 
wheel ; ^'^. 2, of a pillow block : following these examples, let the learner 
take other mechanical details, as given in " Drawing of Machinery," and 
project them in the same way. 

Plate CYIII. is an isometrical projection of a culvert, such as were 
built beneath the Croton Aqueduct, and explains the construction much 
better than by plan and elevation. 

The plates and figures thus ret)resented show the applicability of this 



410 ISOMETRIC AL DKAWING. ^J^ ? / 

method of projection to various examples. Many others might be intro- 
duced, but the principles of this perspective are so easj and intelligible, 
that to multiply examples would be entirely unnecessary. 

Isometrical projection is especially valuable to the mechanical draughts- 
man, embracing as it does the a]3plicability of a scale with pictorial rep- 
resentation. For drawings for the Patent Office it is especially desirable, 
yet when pictorial representation alone is intended, it is not as truthful as 
a drawing in linear perspective. Its office is rather practical than orna- 
mental. 



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